Fakultät Für Mathematik

Fakultät für Mathematik

DISSERTATION

zur Erlangung des akademischen Grades Doctor rerum naturalium (Dr. rer. nat.)

On some Banach Algebra Tools in Operator Theory

vorgelegt von Dipl.-Math. Markus Seidel, geboren am 27. April 1981 in Zwickau, eingereicht am 06. Oktober 2011.

Betreuer: Prof. Dr. Peter Junghanns Prof. Dr. Bernd Silbermann Gutachter: Prof. Dr. Peter Junghanns Prof. Dr. Vladimir S. Rabinovich Seidel, Markus On some Banach Algebra Tools in Operator Theory Dissertation, Fakultät für Mathematik Technische Universität Chemnitz, Oktober 2011 To Gloria, Maria and Emilia. ii Introduction

This work is devoted to sequences A = An of operators An with a specific asymptotic structure. Such sequences usually arise from certain{ } approximation methods applied to bounded linear operators. The fundamental tool in the present approach is to take so-called snapshots W t(A), that are operators which capture some aspects of the asymptotics of the sequence A under consideration and which are derived by a transformation of A and a limiting process. We are particularly interested in criteria for the stability of A. Here, A is called stable if there is an N N such that the operators An are invertible for all n > N and their inverses are uniformly bounded,∈ i.e. sup A−1 < . n>N k n k ∞ We are further interested in the so-called Fredholm property of sequences A = An , a weaker analogon of stability, which provides surprisingly deep connections between the Fredholm{ } proper- t ties of the entries An and certain characteristics of their snapshots W (A), such as their indices, kernels and cokernels. To be a bit more precise, Fredholmness of A means invertibility of its respective coset in a certain quotient algebra, in analogy to the characterization of the operator Fredholm property as invertibility in the Calkin algebra. Moreover, we will derive results on spectral approximation linking the (pseudo)spectra of the An with the (pseudo)spectra of the t −1 snapshots W (A), and describing the convergence of the norms An , A as well as the con- k k k n k dition numbers cond(An). For this, a second essential tool in the present text is given by local principles, which replace the invertibility problem for an element in an algebra by the problem of invertibility of local representatives of this element in a family of (hopefully simpler) local algebras. These Banach algebra techniques and the structure of the outcoming results are not new at all, but possess a long history and have been developed for and applied to many concrete classes of operators and approximation methods. Throughout this text we will try to convey an impression of the diversity of ideas, methods, papers and authors who contributed to their evolution. For the moment we only mention some of the most important milestones: The story has begun with the approach of Silbermann in his groundbreaking paper [83] where the finite section sequences of Toeplitz operators have been studied with the help of two snapshots, and it initiated an evolution which lead to comprehensive monographs such as [12], [56] or [29] among others. In the paper [10] Böttcher, Krupnik and Silbermann collected the local principles of Simonenko, Allan/Douglas and Gohberg/Krupnik, and even presented a framework which permits to relate the norm of an element under consideration to its local norms. The pioneering work on the asymptotics of norms and pseudospectra (at least for the elements in certain sequence algebras which we have in mind) is due to Böttcher [6], [4]. A rather complete and abstract presentation of the whole theory in the case of C∗-algebras, the so-called Standard Model, is subject of the monograph [30] of Hagen, Roch and Silbermann. In this setting the Fredholm theory for operator sequences and the splitting phenomenon of their singular values has been studied. After Böttcher [5] introduced the approximation numbers as substitutes for the singular values into that business in order to extend the observation on the splitting phenomenon iv INTRODUCTION for Toeplitz operators on l2 also to the Banach spaces lp, Rogozhin and Silbermann presented a Banach space based Fredholm theory for matrix sequences [75]. A modification of this approach which is applicable to the finite section sequences of convolution type operators on cones, i.e. to a class of sequences of operators acting on infinite dimensional Banach spaces, was presented by Mascarenhas and Silbermann in [50]. Notice that (almost) all of these variants are based on strong or even -strong 1 convergence, ∗ and many of them are restricted to sequences A of operators An acting on finite dimensional spaces. Particularly, approximation methods for operators on spaces like l∞ stay out of reach. The infinite dimensional applications require additional assumptions like ind An = 0. The aim of the present text is a generalization and unification which covers most versions of the mentioned algebraic framework for structured operator sequences and extends them to the infinite dimensional case (without additional conditions) and to non-strongly converging (so called - strongly converging) sequences. This will put us in a position to close a couple of gaps in the “bigP picture”, and to include more exotic settings (such as operators on spaces with a sup-norm). We prove several results on a much more abstract level than before, such that they become available not only for certain specific classes of operators but help to tackle many applications at one sweep. Actually, the mentioned -strong convergence and the machinery behind this notion will prove to be the key which opens theP door to the generalization and unification we have in mind. Perhaps, after having read this text the reader joins us in believing that this concept is in impressive harmony with the stability- and Fredholm theory, and is rather the natural choice than an unpretentious substitute for the classical notions of strong convergence and Fredholmness. The first part of the thesis is devoted to basic notions and results for bounded linear operators, to their (N, )-pseudospectra, and at its heart we develop the concept of approximate projections . This already appeared in [86] and [73] in elementary forms and was thoroughly revised in [P63]. Here we we look at these results from a different point of view, contribute some new proofs, and answer a long-standing question on the characterization of the -Fredholm property, with important consequences. P The Fredholm theory for sequences A is presented in Part2. Condensed to one sentence the outcome is: Much substantial information on the asymptotics of the entries An of an operator t sequence A = An is stored in the snapshots W (A) and in the Fredholmness of A. In the3rd Part{ we} even achieve a characterization of the Fredholm property of sequences belonging to a certain class of sequence algebras solely in terms of their snapshots. This is done via localization. Throughout the development of the general theory we successively apply the achieved results to band-dominated operators and their finite sections, and conversely we employ this prototypic example as a motivation and a guide through the following steps on the abstract level. Notice that the class of band-dominated operators is subject of many publications [42], [62], [61], [60], [57], [70], [63], [44], [59], [58], [69], [15], mainly done by Rabinovich, Roch, Silbermann and Lindner, and has been the engine for the development of the theory on approximate projections , in a sense. P Finally, Part4 is devoted to several applications. Many of them are well known and have been intensively studied. Nevertheless, having the abstract tools of the previous parts available, we will recover the known results in an impressive conciseness and clarity, and we will be able to harmonize and to extend them into several directions. Parts of the content have already been published or have been submitted for publication in [80],[81], [82] and [78].

1 ∗ Here, (Bn) is said to converge ∗-strongly, if both (Bn) and (Bn) converge strongly. Acknowledgements

I am deeply grateful to my advisors Peter Junghanns and Bernd Silbermann for their unbounded support, their continuous encouragement and the stable environment they provided me during my studies. Their oﬃce doors have always been open for me and I want to thank for the countless conversations that helped to make my ideas converge. vi Contents

1 Bounded linear operators on Banach spaces1 1.1 Preliminaries...... 2 1.1.1 Banach spaces, linear operators and Banach algebras...... 2 1.1.2 Fredholm operators...... 3 1.1.3 Auerbach’s lemma and the versatility of projections...... 4 1.1.4 Notions of convergence...... 6 1.1.5 Example: lp-spaces and the ﬁnite section method...... 7 1.2 Approximate projections and the -setting...... 8 1.2.1 Substitutes for compactness,P Fredholmness and convergence...... 8 1.2.2 On approximation methods for linear equations...... 12 1.2.3 Example: lp-spaces...... 12 1.2.4 The -dichotomy, or how to grasp Fredholmness in the -setting..... 14 1.3 Geometric characteristicsP of bounded linear operators...... P 21 1.3.1 One-sided approximation numbers and their relatives...... 21 1.3.2 Hilbert spaces and lower singular values...... 24 1.4 Spectral properties...... 26 1.4.1 The spectrum...... 26 1.4.2 The (N, )-pseudospectrum...... 26 1.4.3 Convergence of sets...... 32 1.4.4 Quasi-diagonal operators...... 33 1.5 Example: operators on lp-spaces...... 34 1.5.1 Band-dominated operators...... 34 1.5.2 Rich operators...... 37

2 Fredholm sequences and approximation numbers 39 2.1 Sequence algebras...... 40 2.2 T -Fredholm sequences...... 41 J2.2.1 Systems of operators...... 43 t 2.2.2 The approximation numbers of An and the snapshots W An ...... 45 2.2.3 Regular T -Fredholm sequences...... { } 47 2.2.4 The splittingJ property and the index formula...... 48 2.2.5 Stability of sequences A T ...... 50 2.3 A general Fredholm property...... ∈ F 50 2.4 Rich sequences...... 53 2.5 Example: finite sections of band-dominated operators...... 55 2.5.1 Standard finite sections...... 55 2.5.2 Adapted finite sections...... 58 viii CONTENTS

3 Banach algebras and local principles 61 3.1 Central subalgebras, localization and the KMS property...... 62 3.2 Localization in sequence algebras...... 64 3.2.1 Localizable sequences...... 64 3.2.2 Convergence of norms and condition numbers...... 68 3.2.3 Convergence of pseudospectra...... 70 3.2.4 Rich sequences meet localization...... 72 3.2.5 Example: ﬁnite sections of band-dominated operators...... 73 3.3 -algebras...... 76 P3.3.1 Basic deﬁnitions...... 76 3.3.2 Uniform approximate projections and the -Fredholm property...... 77 3.3.3 -centralized elements...... P 79 3.3.4 P-strong convergence...... 80 3.3.5 AP look into the crystal ball...... 82

4 Applications 83 4.1 Quasi-diagonal operators...... 84 4.2 Band-dominated operators on Lp(R,Y ) ...... 87 4.2.1 Standard finite sections...... 87 4.2.2 Adapted finite sections...... 91 4.2.3 Locally compact operators...... 91 4.2.4 Convolution type operators...... 92 4.2.5 A remark on the connection to the discrete case...... 93 4.3 Multi-dimensional convolution type operators...... 94 4.3.1 The operators and their finite sections...... 94 4.3.2 The main result for localizable sequences...... 96 4.3.3 Some members of T ...... 98 4.3.4 Rich sequences...... L 102 4.4 Further relatives of band-dominated operators...... 105 4.4.1 Band-dominated operators on lp(N,X) ...... 105 4.4.2 Slowly oscillating operators...... 106 4.4.3 Toeplitz operators with continuous symbol...... 106 4.4.4 Cross-dominated operators and the flip...... 108 4.4.5 Quasi-triangular operators...... 109 4.4.6 Triangle-dominated and weakly band-dominated operators...... 111 ∞ 4.4.7 Toeplitz operators with symbol in Cp + Hp ...... 112 4.4.8 A larger class of slowly oscillating operators...... 113

Index 115

References 119 Part 1

Bounded linear operators on Banach spaces

In the beginning of this ﬁrst part we recall several elementary deﬁnitions and results, particularly the notions of compactness, Fredholmness and convergence in operator algebras. This is, in large parts, just for the sake of completeness, and the informed reader may skip it. Since these classical notions are too rigid for our purposes, we introduce a fabric of generalizations in Section 1.2, based on so-called approximate projections , and show how to embed the classical setting into this -framework. The main contributions of theP present text to this approach, which was developed inP [86], [73] and [63], are given by Theorems 1.14 and 1.27. Section 1.3 is devoted to the discussion of approximation numbers and their relations to other well known geometric characteristics of linear operators like the singular values and furthermore to the injection and surjection modulus. The latter also play an important role in Section 1.4 where we deal with the spectrum of a bounded linear operator and its approximation via pseudospectra. 2 PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

1.1 Preliminaries 1.1.1 Banach spaces, linear operators and Banach algebras In this text we will exclusively deal with complex Banach spaces, that are normed vector spaces over the field C of complex numbers which are complete with respect to their norms. Therefore we drop this word complex and simply call them Banach spaces. As usual, we denote by N, Z and R the sets of natural, integer and real numbers, respectively. Furthermore, Z+ (Z−) stands for the sets of all non-negative (non-positive) integers. Analogously, we define R+ and R−. For two Banach spaces X, Y the set of all bounded linear operators A : X Y is denoted by (X, Y), can be equipped with pointwise defined linear operations → L (αA + βB)x := αAx + βBx, A, B (X, Y), α, β C, x X ∈ L ∈ ∈ and the so called operator norm

A := sup Ax Y : x X, x X 1 , k kL(X,Y) {k k ∈ k k ≤ } and forms a Banach space, as is well known from every text book on functional analysis. In the case X = Y we shortly write (X) for (X, Y) and note that the composition of two operators, given by (AB)x := A(Bx) withL x X,L turns (X) into a Banach algebra. ∈ L Here a Banach space with another binary operation : , usually called the multiplication in , is saidA to be a Banach algebra if ( , +,· ) formsA × Aa ring → andA the multiplication is related to the normA by the following inequality: A ·

AB A A A B A for all A, B . k k ≤ k k k k ∈ A If there is an element I which fulﬁlls AI = IA = A for every A and which is of norm one then is referred to∈ as A a unital Banach algebra. In this case, an element∈ A A is invertible from the leftA (right) if there is a B with BA = I (AB = I, respectively), and∈ itA is said to be invertible if it is invertible from both∈ A sides. The inverse of an invertible element A is uniquely determined and is usually denoted by A−1. A Banach subalgebra of a unital Banach algebra is said to be inverse closed in if, whenever an element A is invertibleB in , also its inverseAA−1 belongs to . Recall that ifA an element A in a unital Banach∈ B algebra has normA less than one then the NeumannB series Ak converges k∈Z+ in and its sum is the inverse of the element I A. A − P A linear subspace of is called an ideal if AJ and JA for all A and all J . For a closed ideal J of aA Banach algebra , the set∈ J ∈ J ∈ A ∈ J J A / := A + : A A J { J ∈ A} of cosets A + := A + J : J , provided with the operations J { ∈ J } α(A + ) + β(B + ) := (αA + βB) + , (A + ) (B + ) := (AB) + , J J J J · J J for all A, B and α, β C, and the norm ∈ A ∈

A + := inf A + J A : J , k J kA/J {k k ∈ J } is a Banach algebra again, referred to as the quotient algebra of modulo . Clearly, if has a unit element I then I + is a unit in / . Analogously, we getA the quotientJ space X/ZA of a Banach space X modulo aJ closed subspaceA JZ. 1.1. PRELIMINARIES 3

∗ Given a Banach space X, the Banach space X := (X, C) is referred to as the dual space of X and its elements are the so-called linear functionalsL on X. For A (X, Y), the adjoint operator A∗ (Y∗, X∗) is defined by the relation (A∗f)x = f(Ax) for every∈ L f Y∗ and x X. Notice that∈ LA∗ = A for every A (X, Y), and that A is invertible if and∈ only if A∗ is∈ so. k k k k ∈ L Finally, let H be a complex vector space with an inner product , H such that the norm h· ·i x H := x, x X turns H into a complete normed space. Then H is said to be a Hilbert space.k k Thish speciali case of the more general Banach spaces only plays a minor part in this text. Nevertheless,p most ideas and results in the theory which is subject of the forthcoming sections firstly appeared in this much more comfortable setting, and many (or even most) applications involve problems in Hilbert spaces. Therefore we will meet them when talking about the roots, former approaches and important applications. We point out that every element y H defines a bounded linear functional f H∗ by the ∈ y ∈ rule fy(x) := x, y H for every x H. The famous Riesz representation theorem even states h i ∈ ∗ the converse: Every bounded linear functional f H is of the form f = fz with some z H. ∗ ∈ ∈ Moreover, the mapping UH : H H , y fy is an isometric and antilinear bijection. In the sense of this identification→ of Hilbert7→ spaces with their duals one usually introduces the ? Hilbert adjoint A of an operator A (H1, H2) in a slightly different form, namely as the operator A? (H , H ) which fulfills∈ L ∈ L 2 1 ? Ax, y H = x, A y H for all x H and all y H . h i 2 h i 1 ∈ 1 ∈ 2 In other words we have A? = U −1A∗U with A∗ being the adjoint operator as introduced above H1 H2 for the Banach space setting. In all what follows we will denote the norms and inner products without the subscripts indicating the respective spaces, if the situation is unambiguous.

1.1.2 Fredholm operators This section also contains and summarizes material which is well known and can be found (together with proofs) in [25], [12] or in many textbooks on functional analysis. For a bounded linear operator A (X, Y) we deﬁne its kernel, its range and its cokernel by ∈ L ker A := x X : Ax = 0 { ∈ } im A := A(X) = Ax : x X { ∈ } coker A := Y/ im A. The kernel of a bounded linear operator A always forms a closed subspace of X, and its range is a subspace of Y but, in general, it is not necessarily closed. We say that A is normally solvable if im A is a closed subspace of Y. One usually refers to rank A := dim im A as the rank of A. Further, A (X, Y) is called compact if the closure of the set Ax : x X, x 1 in Y is compact. Denote∈ L the set of all compact operators A (X, Y) by{ (X, Y∈). Ink casek ≤ Y}= X we again abbreviate (X, X) by (X). It is well known∈ that L (X) formsK a closed ideal in (X), and the quotient algebraK (X)K/ (X) is usually referred to asK the Calkin algebra. L The adjoint operator A∗ isL compactK if and only if A is compact. Similarly, A∗ is normally solvable if and only if A is so, and in this case dim ker A∗ = dim coker A, dim coker A∗ = dim ker A. Deﬁnition 1.1. An operator A (X, Y) is called Fredholm operator if dim ker A < and dim coker A < . If A is Fredholm∈ thenL the integer ind A := dim ker A dim coker A is referred∞ to as the index∞ of A. − 4 PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

In the following theorems we record the most important properties of the class of Fredholm operators and several equivalent characterizations of the Fredholm property. Actually, the subsequent sections will reveal that these characterizations constitute a corner stone and a guide throughout the whole text. Theorem 1.2. Let Φ(X, Y) denote the set of all Fredholm operators in (X, Y). Then: L Φ(X, Y) is an open subset of (X, Y) and for A Φ(X, Y) we have • L ∈ dim ker(A + C) dim ker A, dim coker(A + C) dim coker A ≤ ≤ whenever C has suﬃciently small norm. The mapping ind : Φ(X, Y) Z is constant on the connected components of Φ(X, Y). → Let A Φ(X, Y) and K (X, Y). Then A + K Φ(X, Y) and ind(A + K) = ind A. • ∈ ∈ K ∈ (Atkinson’s theorem). Let Z be a further Banach space, A Φ(Y, Z) and B Φ(X, Y). • Then we have AB Φ(X, Z) and ind(AB) = ind A + ind B.∈ ∈ ∈ Theorem 1.3. Let A (X, Y). Then the following conditions are equivalent: ∈ L 1. A is Fredholm. 2. A is normally solvable, dim ker A < , and dim ker A∗ < . ∞ ∞ 3. A∗ is Fredholm. 4. There exist projections 1 P (X) and P 0 (Y) s.t. im P = ker A and ker P 0 = im A. ∈ K ∈ K 5. The following is NOT true: For each l N and each > 0 there exists a projection Q (X) or a projection Q0 (Y) with∈ rank Q, rank Q0 l such that AQ < or Q∈0A K< . ∈ K ≥ k k k k In case Y = X the list can be completed by 6. The coset A + (X) is invertible in the Calkin algebra (X)/ (X). K L K Proof. The equivalence of the Assertions 1. till 3. and 6. is well known, and can again be found in the mentioned textbooks. For 4. 1. notice that compact projections are of ﬁnite rank. The role of compact projections is subject⇒ of the next section and the considerations there provide the proof of the remaining implications. In particular, 2. 4., 5. as well as NOT 2. NOT 5., and hence 5. 2., follow from Proposition 1.6. ⇒ ⇒ ⇒ 1.1.3 Auerbach’s lemma and the versatility of projections One of the most valuable instruments in this work is the characterization of subspaces via projections, that are operators P (X) with P 2 = P . ∈ L To be more precise, we say that a closed subspace X1 of a Banach space X is complemented if there is another closed subspace X2 X (a so-called complement of X1) such that X decomposes into X and X in the sense that X⊂= X + X and X X = 0 . In this case, every x X 1 2 1 2 1 ∩ 2 { } ∈ admits a unique decomposition x = x1 + x2 with xi Xi, i = 1, 2. It is well known that the mapping P : x x is a (bounded) projection onto X ∈and I P is a (bounded) projection onto 7→ 1 1 − X2. One further says that P maps parallel to X2. Conversely, for every projection P (X) the space ker P is complemented and im P is a complement. ∈ L

1An operator P ∈ L(X) is called projection if P 2 = P . 1.1. PRELIMINARIES 5

Proposition 1.4. (Auerbach’s lemma 2) n n ∗ Let X be of dimension n. Then there exist xi i=1 X and functionals fi i=1 X such that, for every i, k, { } ⊂ { } ⊂ 1 : i = k xi = fk = 1 and fk(xi) = . k k k k 0 : i = k ( 6 This immediately yields the existence of bounded projections onto ﬁnite dimensional and ﬁnite codimensional subspaces of Banach spaces. For details see [54], B.4.9f.

Proposition 1.5. Let X be a Banach space. If X X is an m-dimensional subspace then 1 ⊂ there is a projection P onto X1 of norm P m. If X2 is a subspace having codimension n then there is a projection P 0 parallel to Xk ofk norm ≤ P 0 n + 1. 2 k k ≤ Now we can capture all Fredholm properties of bounded linear operators in terms of projections, as the next well known proposition shows. For completeness we give its short proof.

Proposition 1.6. Let X, Y be Banach spaces and A (X, Y). ∈ L 1. If A is normally solvable and k dim ker A (k dim coker A) then there is a projection P (X) (P 0 (Y)) of rank k ≤and with norm less≤ than k+2 such that AP = 0 (P 0A = 0, respectively).∈ L ∈ L

2. If A is not normally solvable then, for every k N and every > 0, there are projections P (X) and P 0 (Y) of rank k and with∈ norm less than k + 2 such that AP < and∈ LP 0A < . ∈ L k k k k Proof. In order to prove the second assertion, assume first that im A is not closed. We want to show that for every l N and for every δ > 0 there exists an l-dimensional subspace Xl of X such ∈ that A Xl < δ. Fix δ > 0. Then there exists x1 X, x1 = 1 such that Ax1 < δ, otherwise the rangek | ofk A would be closed. Assume that the∈ assertionk k is true for l 1k. Byk Proposition 1.5 − there exist a complement Z of X − and a projection Q onto X − with ker Q = Z and Q l. l 1 l 1 k k ≤ A(Z) is not closed (otherwise A(X) = A(Xl−1) + A(Z) would be the sum of a finite dimensional and a closed subspace of Y and hence closed, as is known). Now we can choose x Z with l ∈ x = 1 and Ax < δ, and for arbitrary scalars α, β and elements x X − we see k lk k lk ∈ l 1 A(αx + βx ) A(αx) + A(βx ) < δ( αx + βx ) k l k ≤ k k k l k k k k lk αx = Q(αx) = Q(αx + βx ) Q αx + βx k k k k k l k ≤ k kk lk βx = (I Q)(βx ) = (I Q)(αx + βx ) k lk k − l k k − l k (1 + Q ) αx + βx , ≤ k k k lk i.e. A(αx + βx ) < δ(1 + 2l) αx + βx . Since δ > 0 is arbitrary the assertion follows by k l k k lk induction. Consequently, for k N and > 0 we can choose 0 < δ < /k, a k-dimensional subspace of X and an appropriate∈ projection P onto this subspace with P k, such that AP δ P < . k k ≤ Tok findk ≤P 0kink the second assertion we recall that A∗ is not normally solvable whenever A is so, hence there is a projection Q (Y∗) of rank k such that A∗Q < by what we have ∈ L k k k+2 proved before. Let Yˆ be the intersection of the kernels of all functionals in im Q. Obviously, Yˆ has codimension k and we can choose a rank k projection P 0 of norm less than k + 2 (see Proposition 1.5) such that I P 0 is onto Yˆ . Then f (I P 0) = 0 for all f im Q, that is − ◦ − ∈ 2For a proof see [54], B.4.8. 6 PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

(I P 0)∗Q = 0. Since (P 0)∗ and Q are projections of the same (finite) rank we conclude that im(−P 0)∗ = im Q and P 0A = A∗(P 0)∗ = A∗Q(P 0)∗ A∗Q P 0 < . k k k k k k ≤ k kk k Now, for the first assertion apply Proposition 1.5 to find P . To construct P 0, recall that dim ker A∗ and dim coker A coincide in case of a normally solvable operator A, choose a rank-k-projection Q onto a subspace of ker A∗, and proceed as above. Also the notion of generalized invertibility can be completely characterized in terms of projections. Remember that an operator A (X, Y) is said to be generalized invertible, if there is an operator B (Y, X) such that A ∈= LABA and B = BAB. ∈ L Proposition 1.7. Let A (X, Y) and B (Y, X). Then the following conditions are equivalent ∈ L ∈ L

ABA = A. • IX BA is a bounded projection with im(IX BA) = ker A. • − − IY AB is a bounded projection with ker(IY AB) = im A. • − − Moreover, if A (X, Y) is an operator and P (X), P 0 (Y) are projections with im P = ker A and∈ker L P 0 = im A then there exists an∈L operator C∈ L (Y, X) with A = ACA, 0 ∈ L C = CAC and P = IX CA, P = IY AC. − − Proof. Let ABA = A. Then

2 (IX BA) = IX BA BA + BABA = IX BA BA + BA = IX BA and − − − − − − 2 (IY AB) = IY AB AB + ABAB = IY AB AB + AB = IY AB − − − − − − are projections which fulﬁll

im A = im ABA im AB im A, hence im A = im AB = ker(IY AB), and ⊂ ⊂ − ker A ker BA ker ABA = ker A, hence ker A = ker BA = im(IX BA). ⊂ ⊂ − The rest of the first part is obvious. Now let A, P and P 0 be given as stated. Then ker P and ker P 0 are closed. As a consequence 0 of the Banach inverse mapping theorem, the operator A ker P : ker P ker P is invertible. Let A(−1) be its inverse. Then C := (I P )A(−1)(I P 0) does| the job. → − − 1.1.4 Notions of convergence Definition 1.8. A sequence (A ) of operators A (X, Y) is said to converge to A (X, Y) n n ∈ L ∈ L strongly if Anx Ax Y 0 for each x X (we write A = s-lim An), • k − k → ∈ n→∞ uniformly if A A 0. • k n − kL(X,Y) → Let (A ) (X, Y) be a bounded sequence of operators. The Banach Steinhaus Theorem states n ⊂ L that if Anx converges for every x X then Ax := limn Anx defines a bounded linear operator A (X, Y), and ∈ ∈ L A lim inf An . k k ≤ n→∞ k k It is well known that compact operators turn strong convergence into uniform convergence: 1.1. PRELIMINARIES 7

Proposition 1.9. Let X be a Banach space and A, A (X). Then n ∈ L A A strongly A K AK uniformly for every K (X). • n → ⇔ n → ∈ K A∗ A∗ strongly KA KA uniformly for every K (X). • n → ⇒ n → ∈ K ∗ We say that a sequence of operators An converges -strongly, if (An) as well as (An) converge strongly. ∗

1.1.5 Example: lp-spaces and the finite section method Now, it’s time to illustrate the introduced notions and their interactions with an example. Let X stand for a fixed complex Banach space and, for 1 p < , let lp = lp(Z,X) denote the ≤ ∞ 3 Banach space of all sequences x = (x ) ∈ of elements x X such that i i Z i ∈ p p x p := x < . (1.1) k kl k ikX ∞ ∈ Xi Z Note that the linear structure is given by entrywise defined addition and scalar multiplication, and (1.1) defines the norm. We further introduce l∞ = l∞(Z,X) as the Banach space of all x = (xi) with

x l∞ := sup xi X < . k k i∈Z k k ∞ In all what follows we again omit the subscript of the norm if the situation is unambiguous. For 1 p < the dual space of lp(Z,X) can be identiﬁed with lq(Z,X∗) where 1/p + 1/q = 1. Here,≤ the∞ dual product is given by

(x ), (y ) := y (x ). h i i i i i ∈ Xi Z

Let, for a moment, p = 2 and X be a Hilbert space of ﬁnite dimension. Then l2 becomes a separable Hilbert space, provided with the scalar product deﬁned by (x ), (y ) := x , y . i i i∈Z i i X 2 h i h i Moreover, we can introduce projections Pm (l ), m N, by the rule ∈ L ∈ P

P :(x ) (..., 0, x− , . . . , x , 0,...) (1.2) m i 7→ m m and we may easily check that the sequence (Pm)m∈N converges -strongly to the identity. For an equation Ax = b with given A (l2), b l2 and the unknown∗ x l2, the so-called finite ∈ L ∈ ∈ section method uses the (finite dimensional) linear equations PnAPnxn = Pnb as substitutes for Ax = b. One hopes that they provide solutions xn (at least for sufficiently large n) which converge to the desired x. The following result, which is due to Polski [55], justifies this approach.

Proposition 1.10. Let X be a Banach space and suppose that A (X) is invertible, the ∈ L sequence (An) (X) converges strongly to A, (bn) X converges to b X, and (An) is stable, that is ⊂ L ⊂ ∈ −1 N such that An is invertible for n > N and sup An < . ∃ n>N k k ∞

Then Anxn = bn are uniquely solvable for n > N and the solutions xn converge to the (unique) solution x of Ax = b.

3One usually says that the sequence x is p-summable. 8 PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

So, we see that for an invertible A the stability of the ﬁnite section sequence (PnAPn) provides the applicability of the ﬁnite section method to approximate the solution of Ax = b.4 2 It is well known that the projections Pm generate the ideal (l ) of all compact operators in the 2 K 2 sense, that the smallest closed ideal in (l ) which contains all Pm coincides with (l ). This observation conveys an impression that alsoL compact operators will occur in the treatmentK of the stability, and, indeed, it has turned out during the last decades that one has to deal with ideals of “compact” sequences of the form 5

2 (P KP ) + (G ): K (l ), G →∞ 0 . { n n n ∈ K k nk →n }

Now, we want to relax our assumptions on p and X and we ask what happens if 1 p ≤ ≤ ∞ and X is an arbitrary Banach space. The definition of the Pn extends correctly to all cases, the Polski result covers all spaces lp with 1 p < , but p = stays out of reach, because ≤ ∞ ∞ there the sequence (Pn) does not converge strongly anymore. Since the classical approaches for the stability even require -strong convergence in general, also the case p = 1 is unattainable. One reason for this is the∗ fact that in the cases p 1, the ideal which is generated by the ∈ { ∞} operators Pn does not coincide with the ideal of the compact operators anymore. Moreover, since the projections Pn are non-compact if dim X = , the picture also gets dramatically worse in the infinite dimensional case. ∞ Therefore, the next section is devoted to a slightly modified framework, which will again provide certain notions of compactness, Fredholmness and convergence having similar properties and interactions, which will allow to restate and to extend the Polski result and the theory which is necessary to characterize the stability, but which is more flexible. In particular, we aspire to an equal treatment of all cases 1 p and all spaces X. ≤ ≤ ∞ 1.2 Approximate projections and the -setting P Looking back at the development and also at the modern proofs in the Banach algebra techniques which are used for the theory of approximation methods it becomes apparent that the interactions between compactness, Fredholmness, and strong convergence6 constitute a large part of the core. We now want to modify this triple such that these fruitful relations survive and in this way also the familiar proof strategies still work, but a larger flexibility is achieved, such that we can also cover approximation methods which do not converge strongly. In a way, we turn the table and we no longer hunt for methods which suit to the classical notions of compactness, Fredholmness and convergence, but we take the method itself as the starting point and construct a suitable triple.

1.2.1 Substitutes for compactness, Fredholmness and convergence

Approximate projections Let X be a Banach space and let = (Pn)n∈N be a bounded sequence of operators in (X) with the following properties: P L Pn = 0 and Pn = I for all n N, • 6 6 ∈ For every m N there is an Nm N such that PnPm = PmPn = Pm if n Nm. • ∈ ∈ ≥ 4 p Notice that for an operator A ∈ L(l ) the ﬁnite section sequence (PnAPn) is stable (where the operators p PnAPn are considered as operators in L(im Pn), respectively) if and only if the sequence (An) ⊂ L(l ) with An = PnAPn + (I − Pn) is stable. 5See, for example, the pioneering paper [83] of Silbermann, the monograph [12] or our applications in Part4. 6See Theorem 1.3 and Proposition 1.9 1.2. APPROXIMATE PROJECTIONS AND THE -SETTING 9 P

7 Then is referred to as an approximate projection. In all what follows we set Qn := I Pn and weP write m n if P Q = Q P = 0 for all k m and all l n. − k l l k ≤ ≥ -compactness Let be an approximate projection. A bounded linear operator K is called P P -compact if KPn K and PnK K tend to zero as n . By (X, ) we denote the setP of all -compactk − operatorsk k on X and− k by (X, ) the set→ of all∞ boundedK linearP operators A for whichPAK and KA are -compact wheneverL KPis -compact. P P Theorem 1.11. Let be an approximate projection on the Banach space X. (X, ) is a closed subalgebra of (PX), it contains the identity operator, and (X, ) is a properL closedP ideal L K P of (X, ). An operator A (X) belongs to (X, ) if and only if, for every k N, L P ∈ L L P ∈ P AQ 0 and Q AP 0 as n . k k nk → k n kk → → ∞ The “ -concept” which we are introducing here will be developed later on in Part3 also in more abstractP Banach algebras. Therefore we will elaborate the details there and, for the moment, we refer to Theorem 3.25.

-Fredholmness and invertibility at inﬁnity Here are two possible generalizations of Fred- Pholmness based on -compact operators instead of compact ones which are motivated by Theo- rem 1.3. P

Definition 1.12. We say that A (X) is invertible at infinity (with respect to ) if there is an operator B (X) with I ∈AB, L I BA (X, ). In this case B is referredP to as a -regularizer for A∈. L − − ∈ K P PAn operator A (X, ) is said to be -Fredholm if the coset A + (X, ) is invertible in the quotient algebra∈ L(X, P)/ (X, ). P K P L P K P Notice that invertibility at infinity is defined in (X), whereas for -Fredholmness we are restricted to (X, ), since we need that (X, ) Lforms a closed idealP in (X, ). Both notions have been knownL P and have been studiedK for aP long time. The monographsL by Rabinovich,P Roch and Silbermann [63] and Lindner [44] already contain a comprehensive theory and many applications of this approach. But until now it was an open problem if these two notions coincide for operators A (X, ). Here we answer this question affirmatively, under a natural condition on which has∈ L beenP in the business since the inverse closedness of (X, ) in (X) is known, basedP on a proof of Simonenko [86]. L P L

Definition 1.13. Given a Banach space X with an approximate projection = (P ), we set P n S1 := P1 and Sn := Pn Pn−1 for n > 1. Further, for every finite subset U N, we define − ⊂ P := S . is said to be uniform if CP := sup P < , the supremum over all finite U k∈U k P k U k ∞ U N. ⊂ P Theorem 1.14. Let be a uniform approximate projection on X. An operator A (X, ) is -Fredholm if and onlyP if it is invertible at infinity. In this case every -regularizer∈ of LA belongsP toP (X, ). Particularly, (X, ) is inverse closed in (X). P L P L P L This main result (except the inverse closedness) is really new and due to the author, closes a long-standing gap in the theory of approximate projections and has been published in [81] for the first time. It is a special case of Theorem 3.28 whose proof is essentially based on Theorem 3.27 which is new as well, interesting on its own and, in the present context, reads as follows.

7This notion was introduced in [73]. 10 PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

Theorem 1.15. Let be a uniform approximate projection on X and A (X, ). Then there P ˆ ∈ L P is an equivalent uniform approximate projection = (Fn) on X (depending on A) with CPˆ CP such that P ≤ [F ,A] = AF F A 0 as n . k n k k n − n k → → ∞ Here, two approximate projections = (P ) and ˆ = (F ) on X are said to be equivalent if for P n P n every m N there is an n N such that ∈ ∈ PmFn = FnPm = Pm and FmPn = PnFm = Fm. Note that in this case we have (X, ) = (X, ˆ). Indeed, for K (X, ), K P K P ∈ K P K(I F ) KQ I F + K P (I F ) , k − n k ≤ k mkk − nk k kk m − n k where the first summand tends to zero as m and, for every fixed m, the second one equals zero for an n by the definition, and then→ also ∞ for all n n . Analogously, we check 0 Pˆ 0 (I Fn)K 0 as n , hence K (X, ˆ). Reversing the roles of and ˆ gives kthe− claim. Consequently,k → → also ∞ (X, ) = ∈( KX, ˆ)Pand the notions of -compactnessP P and - Fredholmness coincide with theL respectiveP Lˆ-notions.P The same holdsP true for the so-calledP -strong convergence, a substitute for the thirdP part of the triple, the strong convergence, which weP introduce in the next paragraph.

-strong convergence Let = (Pn) be an approximate projection and, for each n N, let P P ∈ An (X). The sequence (An) converges -strongly to A (X) if, for all K (X, ), both (A∈ L A)K and K(A A) tend to 0Pas n . In this∈ L case we write A ∈ KA -stronglyP k n − k k n − k → ∞ n → P or A = -limn An. We mention that this definition is based on the similar properties of strong convergenceP as stated in Proposition 1.9 and was introduced in [73]. Proposition 1.16. A bounded sequence (A ) in (X) converges -strongly to A (X) iff n L P ∈ L (A A)P 0 and P (A A) 0 for every fixed P . (1.3) k n − mk → k m n − k → m ∈ P This easily follows from (X, ) and the estimates P ⊂ K P (A A)K K (A A)P + A A Q K k n − k ≤ k kk n − mk k n − kk m k K(A A) K P (A A) + A A KQ . k n − k ≤ k kk m n − k k n − kk mk Unfortunately, the -strong limit is not unique in general. Consider, for instance, a projection P P/ 0,I and = (Pn) given by Pn := P for all n. Then the sequence (Pn) converges -strongly∈ { } to bothP P and I. Therefore we adopt a further condition on from [63] to force uniqueness.P P Definition 1.17. An approximate projection is called approximate identity if for every x X the estimate sup P x x holds. P ∈ n k n k ≥ k k By (X, ) we denote the set of all bounded sequences (An) (X), which possess a -strong limitF in P(X, ). ⊂ L P L P Remark 1.18. Let be an approximate identity. Then A sup P A holds for every P k k ≤ m k m k bounded linear operator A. To check this, fix > 0 and choose x0 X, x0 = 1 such that Ax A . Furthermore, take n such that P Ax Ax ∈ tok getk k 0k ≥ k k − 0 k n0 0k ≥ k 0k −

A Pn0 Ax0 + 2 Pn0 A + 2 sup PmA + 2. k k ≤ k k ≤ k k ≤ m∈N k k This gives the claim, since was chosen arbitrarily. 1.2. APPROXIMATE PROJECTIONS AND THE -SETTING 11 P

Besides the uniqueness of -strong limits this provides several further structural properties. P Theorem 1.19. Let be an approximate identity on X. Then 8 P The algebra (X, ) is closed with respect to -strong convergence, this means if a sequence • L P P (An) (X, ) converges -strongly to A then A (X, ). Moreover, (An) is bounded in this⊂ case, L andP hence belongsP to (X, ). ∈ L P F P The -strong limit of every (A ) (X, ) is uniquely determined. • P n ∈ F P Provided with the operations α(A ) + β(B ) := (αA + βB ), (A )(B ) := (A B ), and • n n n n n n n n the norm (An) := supn An , (X, ) becomes a Banach algebra with identity I := (I). The mappingk (kX, ) k (Xk, F) whichP sends (A ) to its limit A = -lim A is a unital F P → L P n P n n algebra homomorphism and (with BP := sup P ) n k nk

A BP lim inf An . (1.4) k k ≤ n→∞ k k

Proof. For the details of the proof we again refer to the more general result given with Theo- rem 3.34 and note that we can set P := P due to Remark 1.18 and Remark 3.33. D B

Some remarks on inclusions It is obvious that for arbitrary Banach spaces X with an approximate projection the inclusions P (X, ) ( (X, ) (X) K P L P ⊂ L hold. To disclose how the set of compact operators ﬁts into that picture we borrow the following result from [44], Proposition 1.24.

Proposition 1.20. The inclusion (X, ) (X) holds if and only if (X). If is an approximate identity on KX thenP we⊂ alsoK have P ⊂ K P (X, ) (X) (X, ). L P ∩ K ⊂ K P Proof. At first, recall that every operator A (X, ) is the norm limit of the sequence (AP ). ∈ K P n Thus it is compact if all APn are compact. If, conversely, one Pm is not compact, then it belongs to (X, ) (X). K P \K Now let be an approximate identity, let A (X, ) be compact, and assume that AQn does notP tend to zero. Then there is a bounded∈ LsequenceP (x ) X such that AQ x doesk notk n ⊂ n n tend to zero as well, but with the help of Theorem 1.11 we see that at least PkAQnxn 0 as n for every fixed k. Since A is compact, it is always true that each subsequence of (AQ→ x ) → ∞ n n has a convergent subsequence and the respective limit y fulfills Pky = 0 for every k, hence y = 0 since is an approximate identity. This contradicts AQ x 0. P n n 6→ To show that also Q A 0, we mention that k n k → Q A Q AP + Q AQ , k n k ≤ k n kk k nkk kk where the last term at the right hand side can be made as small as desired by choosing k large and the first one tends to zero for every fixed k by Theorem 1.11.

8These results are already known from [63], Section 1.1.4 et seq. 12 PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

1.2.2 On approximation methods for linear equations Let X be a Banach space with an approximate projection . In accordance with the definition P for operator sequences we say that a sequence (xn) of elements xn X converges -strongly to x X if ∈ P ∈ K(x x) 0 as n , for every K (X, ). k n − k → → ∞ ∈ K P It is obvious from the definition that a bounded sequence (x ) X converges -strongly, iff n ⊂ P

Pm(xn x) 0 as n , for every m N. k − k → → ∞ ∈ Deﬁnition 1.21. A Banach space X with an approximate projection is called complete w.r.t. -strong convergence if the following holds: Every bounded sequenceP (x ) X for which P n ⊂ (Pmxn)n∈ is a Cauchy sequence for every m N has a -strong limit x X. N ∈ P ∈ Here comes the announced generalized version of the Polski result (Proposition 1.10) which was discovered by Roch and Silbermann in [73]. Proposition 1.22. Let be a uniform approximate projection on the Banach space X. Suppose that (b ) X is boundedP and converges -strongly to b X, A (X, ) is invertible, and n ⊂ P ∈ ∈ L P the sequence (An) (X) is stable and converges -strongly to A. Then the solutions xn of A x = b converge⊂ L-strongly to the solution x of AxP = b. n n n P Proof. Let K (X, ), n be suﬃciently large such that A is invertible, and consider ∈ K P n K(x x) = K(A−1b A−1b) = K((A−1 A−1)b + A−1(b b)) k n − k k n n − k k n − n n − k = KA−1(A A )A−1b + KA−1(b b) k − n n n n − k KA−1(A A ) A−1b + KA−1(b b) . ≤ k − n kk n nk k n − k The operator A−1 belongs to (X, ) by Theorem 1.14, hence KA−1 is -compact. This obvi- −1 L P −1 P −1 ously yields that KA (A An) and KA (bn b) tend to zero as n . Since An bn are uniformly boundedk w.r.t.− n, thek assertionk follows.− k → ∞ k k

1.2.3 Example: lp-spaces Let us again rest for a minute and ask, what this means for the spaces lp = lp(Z,X) which were introduced above in Section 1.1.5 and for the finite section method applied to bounded linear operators on these spaces. First of all we note that the sequence = (P ) as defined in (1.2) gives a uniform approximate P n identity with BP = CP = 1 for all 1 p and all Banach spaces X. Therefore all results of Section 1.2 can be applied. That is,≤ the≤ algebra ∞ (lp, ) is closed under passing to inverses or -regularizers (by Theorem 1.14), the notions “ L-Fredholm”P and “invertible at infinity” are equivalent,P and -strong convergence is well definedP (see Theorem 1.19). We will even see that Fredholm operatorsP in (lp, ) are automatically -Fredholm (Corollary 1.28). Figure 1.1 illustrates theL relationsP between the respectiveP operator algebras (lp), (lp), (lp, ), and (lp, ) for the various configurations of p and dim X. To substantiateL theK assertedL properP inclusionsK P there, we consider two examples which we again take from [44].

Let a X and f X∗ be non-zero elements and let A (l1) be given by the rule • ∈ ∈ ∈ L

A :(x ) ..., 0, 0, f(x )a, 0, 0,... . i 7→ i i ! X 1.2. APPROXIMATE PROJECTIONS AND THE -SETTING 13 P

dim X < dim X = ∞ ∞

(lp) = (lp, ) (lp) .B L L P L I I . (lp, ) . L P

(lp, ) .Pn K P 1 < p < ∞ (lp) = (lp) K K (lp, ) K P .Pn

(lp) I+A (lp) B I+A L . L . . (lp, ) .I (lp, ) .I L P L P

(lp, ) .Pn p = 1, K P p = ∞ .A .A (lp, ) K P .Pn (lp) (lp) K K

Figure 1.1: Venn diagrams of (lp), (lp), (lp, ), and (lp, ) depending on lp = lp(Z,X). L K L P K P

This operator is compact, but does not belong to (l1, ) due to Theorem 1.11. The adjoint of A provides the same outcome for the case Lp = P. ∞ Let X be the space Lp[0, 1] of all p-Lebesgue integrable functions over the interval [0, 1] • ˜ p and deﬁne B : l (Z,X) X, (ui) v by → 7→ 1 1 uk(x): if x 1 k 1 , 1 k for one k N v(x) := ∈ − 2 − − 2 ∈ otherwise. (0 : (As customary we let L∞[0, 1] stand for the space of all Lebesgue measurable functions that are essentially bounded.) Then the linear operator B :(u ) ..., 0, 0, B˜(u ), 0, 0,... i 7→ i acts boundedly on lp = lp(Z,X) for every p, but does not belong to (lp, ) in any case. L P p For an operator A (l , ) the ﬁnite section sequence (PnAPn) is stable (where the operators P AP are considered∈ L as operatorsP in (im P ), respectively) if and only if (A ) (lp, ) with n n L n n ⊂ L P 14 PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

p An = PnAPn + Qn is a stable sequence. Moreover, for b l , the sequence (Pnb) is always bounded and converges -strongly to b. Thus, Proposition 1.22∈ , applied to this concrete setting, reads as follows. P Corollary 1.23. Let A (lp, ) be invertible and its finite section sequence be stable. Then ∈ L P the solutions xn of the truncated equations PnAPnxn = Pnb exist for sufficiently large n and they converge -strongly to the solution x of the equation Ax = b.9 P 1.2.4 The -dichotomy, or how to grasp Fredholmness in the -setting P P From Theorem 1.3 we know that the usual Fredholm property of bounded linear operators can also be described in terms of compact projections. Here comes the analogon based on -compact projections which was firstly studied in [80] and refined to the present shape in [81]. P Definition 1.24. An operator A (X) is said to be properly -Fredholm, if there exist projections P,P 0 (X, ) such that∈ Lim P = ker A and ker P 0 = im PA. An operator A ∈( KX) isP called properly -deficient from the right (left) if, for each > 0 and ∈ L P each k N, there is a projection R (X, ) of rank at least k such that AR < ( RA < , respectively).∈ ∈ K P k k k k Of course, it would be a great benefit to capture the usual Fredholm property (and the related attributes like the dimensions of the kernel and cokernel, the index, or the normal solvability) by -compact projections. Concerning this let us first get a short overview, the proofs and a more detailedP discussion will follow afterwards. We initially condense our aim to a definition and give a first result. Definition 1.25. An operator A (X) is said to have the -dichotomy if it is either Fredholm and properly -Fredholm, or it is∈ properly L -deficient fromP at least one side. Further, we sayP that X has the -dichotomy,P if every operator A (X, ) has the -dichotomy. P ∈ L P P Proposition 1.26. Let be an approximate projection and let A (X) be invertible at infinity. Then A has the -dichotomy.P ∈ L P Notice that, in case rank Pn < for all n, every -compact operator is compact. Hence, invertibility at infinity implies Fredholmness,∞ and byP the previous Proposition also proper - P Fredholmness. If, on the other hand, not all Pn are of finite rank then there are operators which are not Fredholm but invertible at infinity and hence properly -deficient. A quite trivial 2 P example is the operator A = Q1 on l (Z,X) with dim X = . Nevertheless, we need practicable criteria to ensure this -dichotomy∞ for A, even if we do not know anything about its invertibility at infinity. As inP the preceding sections also here the situation becomes much more comfortable if we tighten the assumptions on . P Theorem 1.27. Let = (Pn) be a uniform approximate identity on the Banach space X. Then X has the -dichotomyP if one of the following conditions is fulfilled: P ∗ = (P ∗) is an approximate identity on X∗. •P n X is complete w.r.t. -strong convergence. • P With these sufficient conditions we have everything what we (and the reader) need to know about the -dichotomy to cope with the material in the forthcoming sections. In particular, for our prototypicP example lp we get 9Actually, the demand on the invertibility of A is even redundant since it follows from the stability as we will see in the next step. 1.2. APPROXIMATE PROJECTIONS AND THE -SETTING 15 P

Corollary 1.28. All lp-spaces have the -dichotomy. P Proof. With the identiﬁcations (lp(Z,X))∗ = lq(Z,X∗) for 1 < p < and 1/p + 1/q = 1, as ∞ well as (l1(Z,X))∗ = l∞(Z,X∗) we easily deduce that ∗ is an approximate identity in all the cases except for p = . Furthermore, the spaces l∞ areP complete w.r.t. -strong convergence. Thus, Theorem 1.27 gives∞ the claim. P

Moreover, we mention that, for every operator A (lp, ), the stability of its ﬁnite section ∈ L P sequence (PnAPn) already implies the invertibility of A, as the following Proposition shows. This simpliﬁes Corollary 1.23.

Proposition 1.29. Let be a uniform approximate identity, (An) (X, ) be stable, and P ∈ F P −1 A := -limn An have the -dichotomy. Then A is invertible and the inverses An converge -stronglyP to A−1. P P Proof. For large n and every K (X, ) ∈ K P −1 An 1 AnK = k −1k AnK −1 K . k k An k k ≥ An k k k k k k For n , we obtain AK C K and analogously KA C K for every -compact → ∞ k−1 k−1 ≥ k k k k ≥ k k P K, where C (supn An ) > 0 is constant. Thus, A is not -deficient, hence properly -Fredholm.≥ Supposek thatk the kernel of A is not trivial. Then thereP is a non-trivial projection P (X, ),P = 0 such that 0 = AP C P C, yielding a contradiction. Thus A is injective.∈ K AnalogouslyP 6 one shows thatk A kis ≥ surjectivek k ≥ and hence invertible, due to the Banach inverse mapping theorem. From Theorem 1.14 we even get A−1 (X, ). Further, we know for large n and every K (X, ) that ∈ L P ∈ K P (A−1 A−1)K = A−1(I A A−1)K = A−1(A A )A−1K, n − n − n n − n which obviously converges to zero in the norm as n . In the same way we find that K(A−1 A−1) 0 in the norm, hence the -strong convergence→ ∞ A−1 A−1 follows. n − → P n → Proofs and technical details Let us try to gather more understanding of the somewhat mysterious -dichotomy. First of all, from Proposition 1.7 we immediately conclude P Corollary 1.30. An operator A (X) is properly -Fredholm iff there is a -regularizer B (X) for A which is also a generalized∈ L inverse, thatP is I AB, I BA P(X, ) and ABA∈ L= A, BAB = B. − − ∈ K P If A (X, ) and is even uniform then the situation becomes more relaxed. ∈ L P P Corollary 1.31. Let be a uniform approximate projection. Then A (X, ) is properly -Fredholm if and onlyP if it is -Fredholm and has a generalized inverse∈ in L (XP, ). In this Pcase there is a B (X, ) whichP is -regularizer and generalized inverse at theL sameP time. ∈ L P P Proof. Let A (X, ) be properly -Fredholm. Then, by Corollary 1.30, there is an operator B (X) such∈ Lthat AP= ABA, B = PBAB and I AB, I BA (X, ). From Theorem 1.14 we∈ obtain L that B (X, ). − − ∈ K P Conversely, let A ∈be L -FredholmP with -regularizer C (X, ) and let B (X, ) be a generalized inverse forPA. The projectionsP P := I BA, P∈0 := L I PAB are contained∈ L in P(X, ). Moreover, P = (I CA)P + CAP = (I CA)P −and P 0 = P 0(I− AC) + P 0AC = P 0(LI ACP ) even show that P,P− 0 (X, ), that is−B is a -regularizer for−A. In view of Corollary− 1.30, this yields the proper ∈-Fredholmness. K P P P 16 PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

Proposition 1.26 immediately results from the following observation. Proposition 1.32. Let A (X) be invertible at infinity (w.r.t. an approximate projection ). ∈ L P If A is normally solvable then for every k N with k dim ker A (k dim coker A) there is a -compact projection P of rank k such that∈AP = 0 (PA≤ = 0). ≤ IfP A is not normally solvable then A is properly -deficient from both sides. P Proof. Let B (X) be a -regularizer for the operator A. For every x ker A we find that Q x = Q (I ∈BA L )x + Q BAxP = Q (I BA)x tends to zero as n ∈. Let X be a finite n n − n n − → ∞ 1 dimensional subspace of ker A. Then we can fix m N s.t. sup Qmx : x X1, x = 1 < 1/2 and deduce that the operator P : X X := P ∈(X ) is invertible{k andk its∈ inversek hask norm} less m 1 → 2 m 1 than 2. Let S denote its inverse and let m˜ m. Since X2 is a finite dimensional subspace of the Banach space ker Q , there is a bounded projection R (ker Q ) onto X with R dim X , m˜ ∈ L m˜ 2 k k ≤ 1 by Proposition 1.5. Now, we define P := SRPm and we easily check that im P = X1 as well 2 as P = SRPmP = SPmP = P , hence P is a projection onto X1 which obviously belongs to (X, ). K P Assume now that A is not normally solvable and fix > 0 and k N. Proposition 1.6 provides us with a rank-k-projection T such that AT . Further, denote∈ by d the finite number sup Q and check that k k ≤ n k nk Q T Q (I BA)T + Q BAT k n k ≤ k n − k k n k Q (I BA) T + d B 2d B ≤ k n − kk k k k ≤ k k for sufficiently large n. Further, for x im T , ∈ AP x Ax + A Q x 1 + 2d B A k n k k k k kk n k k kk k . P x ≤ x Q x ≤ 1 2d B k n k k k − k n k − k k This shows that for sufficiently small and sufficiently large l the space X3 := im PlT is of dimension k and Az 4d B A z holds for all z X . Since X ker Q˜ ker Q for k k ≤ k kk k k k ∈ 3 3 ⊂ l ⊂ ˆl ˆl ˜l l we can again choose a projection R (ker Q ) onto X with norm at most k and ∈ L ˆl 3 define P := RP˜l. Obviously, P is a -compact projection of rank k and we have the estimate AP 4kd(d + 1) B A . Since Pwas chosen arbitrarily, this proves the proper -deficiency fromk k the ≤ right. k kk k P Further, if A is not normally solvable, then A∗ is not normally solvable, too, and for every prescribed > 0 we find a projection T (X∗, ∗) of rank k with A∗T . We show that there is a projection P 0 (X, ) of norm∈ K less thanP 2k(k + 1)(d + 1)ksuch thatk ≤ im(P 0)∗ = im T , and hence ∈ K P P 0A = A∗(P 0)∗ = A∗T (P 0)∗ 2k(k + 1)(d + 1). k k k k k k ≤ Then, since was chosen arbitrarily, A proves to be properly -deficient from the left. For this, let l be such that Q∗T < 1/2. Then, for every f im T weP have k l k ∈ 1 f P = P ∗f f Q∗f = f (Q∗T )f f , (1.5) k ◦ lk k l k ≥ k k − k l k k k − k l k ≥ 2k k that is f : f im T forms a linear space of dimension k. Hence, for ˜l l, { |im Pl ∈ }

X1 := ker f ker Q ker Q˜ | l˜ ⊂ l ∈ f \im T has codimension k in ker Q˜l. Due to Proposition 1.5 we can choose a projection R parallel to X1 onto a certain complement X2 of X1 in ker Q˜l with the norm not greater than k + 1. Since the 1.2. APPROXIMATE PROJECTIONS AND THE -SETTING 17 P

set of all restrictions g = f X of the functionals f im T to X forms a k-dimensional space we | 2 ∈ 2 conclude that each functional on X2 is of the form g = f X2 with f im T . Auerbach’s Lemma (Proposition 1.4) provides bases x , . . . , x X and f , . .| . , f im T∈ such that x = g = 1 1 k ∈ 2 1 k ∈ k ik k jk and gj(xi) = δij for all i, j = 1, . . . , k, where gj := fj X2 (here δij = 1 if i = j, and δij = 0 otherwise). Due to (1.5) the norms f can be estimated| by k jk f 2 f P 2 f R P + 2 f (I R) P 2 g R P < 2(k + 1)(d + 1), k jk ≤ k j ◦ lk ≤ k j ◦ ◦ lk k j ◦ − ◦ lk ≤ k jkk kk lk thus, defining P 0x := k f (x)x , we obtain a -compact projection onto X with the norm i=1 i i P 2 less than 2k(k + 1)(d + 1). Since f (P 0x) = k f (x)f (x ) = f (x) for all j and x we find that P j i=1 i j i j im(P 0)∗ = im T . It remains to consider operators A which areP normally solvable, hence dim coker A = dim ker A∗, and to check that for every finite k dim coker A there is a -compact projection P 0 of rank k with P 0A = 0. Since A∗ is invertible≤ at infinity (w.r.t. ∗)P we can apply the first part of this proof to find a ∗-compact projection T of rank k ontoP a respective subspace of ker A∗. Then we simply applyP the above argument with = 0 to construct P 0. For the proof of Theorem 1.27 we need some preliminary results and we introduce a useful and important closed subspace X0 of X by X := x X : Q x 0 as n . 0 { ∈ k n k → → ∞} Proposition 1.33. .

1. For each A (X, ) the restriction A X to X is contained in (X , ) and if K ∈ L P | 0 0 L 0 P ∈ (X, ) then K X (X , ). K P | 0 ∈ K 0 P 2. Let X X be a ﬁnite dimensional subspace. Then there is a projection P (X, ) 1 ⊂ 0 ∈ K P onto X with the norm not greater than 2BP dim X , where BP := sup P . 1 1 n k nk 3. If = (P ) is an approximate identity on X and A (X, ) then P n ∈ L P −2 B A A X A , (1.6) P k kL(X) ≤ k | 0 kL(X0) ≤ k kL(X)

Furthermore, for every T (X0, ) there is a lifting K (X, ) of T , that is K X0 = T . Restrictions and liftings of∈ K-compactP projections are again∈ K projectionsP of the same| rank. P Proof. 1. Let A (X, ) and x X0. Since QnAx QnAPl x + Qn A Qlx , where the latter term∈ is L smallerP than any∈ prescribedk > 0k ≤if kl is largekk enough,k k andkk kk the firstk term tends to zero for any fixed l and n , we find that Ax X . Hence A X (X , ). If → ∞ ∈ 0 | 0 ∈ L 0 P K (X, ) then it is also clear by the definition that K X (X , ). ∈ K P | 0 ∈ K 0 P 2. Recall the projection P := SRPm from the proof of Proposition 1.32 which obviously fulfills P S R P 2BP dim im R. k k ≤ k kk kk mk ≤ 3. For every > 0 there is an m N such that A PmA + (see Remark 1.18) and for all sufficiently large n we have P AQ∈ . Nowk (1.6k ≤) easily k followsk by k m nk ≤ 2 A X A P AP + 2 B A X + 2. k | 0 k ≤ k k ≤ k m nk ≤ P k | 0 k

Further, let T (X0, ). From T PnTPn L(X0) 0 as n we conclude for the sequence (P TP ) (∈X K, ) andP n, m largek − that k → → ∞ n n ⊂ K P P TP P TP B2 P TP P TP k n n − m mkL(X) ≤ P k n n − m mkL(X0) 2 B ( P TP P + T P TP ) →∞ 0. ≤ P k n n − kL(X0) k − m mkL(X0) →n,m Hence (P TP ) (X, ) is a Cauchy sequence. For its limit K (X, ) we easily check n n ⊂ K P ∈ K P that the compression K X coincides with T . The rest is easy to prove. | 0 18 PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

Here now comes an extended version of Theorem 1.27.

Theorem 1.27*. Let be a uniform approximate identity. Then A (X, ) has the - dichotomy if one of theP following conditions is fulﬁlled: ∈ L P P

∗ is an approximate identity on X∗. •P There are a Banach space Y and operators R ,B (Y) such that Y∗ = X, R∗ = P for • n ∈ L n n all n N and B∗ = A. ∈

For every properly -Fredholm operator B (X0, ) the sequence (BPn) has a -strong • limit in (X). P ∈ L P P L X is complete w.r.t. -strong convergence. • P Proof. We prepare the proof with some basic observations in the first and a technical result in the second step. 1st step. Let ˆ = (F ) be a uniform approximate projection given by Theorem 1.15. Notice first P n that for every fixed k and sufficiently large n, Pkx = PkFnx CP Fnx . Hence, we deduce that k k k k ≤ k k −1 lim sup Fnx CP x for all x X. (1.7) n→∞ k k ≥ k k ∈

Moreover, for each x X we have Qnx 0 if and only if (I Fn)x 0 as n . Further we write m n if F∈ (I F ) = (kI Fk) →F = 0 for all k km −and allkl → n. → ∞ Pˆ k − l − l k ≤ ≥ 2nd step. Suppose that, for given , δ > 0 and x X with x = 1, we have Ax and lim sup (I F )x δ. Then there is an integer ∈N such that,k k for all n, m Nk , k ≤ n k − n k ≥ 0 ≥ 0

AF x [A, F ] + F Ax 2CP , hence A(F F )x 4CP . k n k ≤ k n k k nkk k ≤ k n − m k ≤ Choose n N such that (I F )x δ/2 and m n such that, due to (1.7), 1 Pˆ 0 k − n1 k ≥ 1 Pˆ 1 −1 −1 (F F )x = F (I F )x (2CP ) (I F )x (4CP ) δ. k m1 − n1 k k m1 − n1 k ≥ k − n1 k ≥

(Fm1 −Fn1 )x Set x1 := k(F −F )xk and ﬁx N1 Pˆ m1. m1 n1 We iterate this procedure as follows: Suppose that n , m ,N , x for k = 1, . . . , l 1 are given. k k k k − Then we analogously choose nl Pˆ Nl−1 such that (I Fnl )x δ/2, and ml Pˆ nl such that (F −F k)x − k ≥ −1 , and set ml nl as well as . By doing this, we (Fml Fnl )x (4CP ) δ xl := k(F −F )xk Nl Pˆ ml k − k ≥ ml nl obtain a set x1, . . . , xl X and integers N0,...,Nl such that (FNk FNk 1 )xi = δikxi. For l { } ⊂ − − y = i=1 αixi and every k = 1, . . . , l we ﬁnd

P l l 1 1 1 y = αixi (FNk FNk 1 ) αixi αkxk = αk − k k ≥ FNk FNk 1 − ≥ CP k k CP | | i=1 k − − k i=1 X X and hence l l 3 4CP 16CP l Ay α Ax CP y 4CP = y . k k ≤ | i|k ik ≤ k k δ δ k k i=1 i=1 X X Let N N and choose a projection R (ker(I F )) of norm at most l onto l+1 Pˆ l ∈ L − Nl+1 span x1, . . . , xl ker(I FNl ) ker(I FNl+1 ), due to Proposition 1.5. Then P := RFNl is a { } ⊂ − ⊂16C4 l2 − projection of rank l and AP P . Moreover, P is ˆ-compact, hence -compact. k k ≤ δ P P 1.2. APPROXIMATE PROJECTIONS AND THE -SETTING 19 P

3rd step. If there is an x ker A such that the norms Qnx do not tend to zero then the second step yields that for every∈k and every γ > 0 there isk a -compactk projection P of rank k such that AP < γ, hence A is properly -deficient from theP right. k k P If, on the other hand, Qnx 0 as n for all x ker A, i.e. ker A X0 then, due to Proposition 1.33, 2.k we havek → that for→ Fredholm ∞ operators∈ A there is always⊂ a -compact projection onto its kernel, and if A has an infinite dimensional kernel then it is properlyP - deficient. P

4th step. Suppose that A X0 is not normally solvable, that is for every fixed > 0 and every k N | ∈ there is a subspace X1 X0 of dimension k such that A X1 /(2kBP ) (see Proposition 1.6). By Proposition 1.33, 2.⊂ we can choose a projection Pk | k(X ≤, ) onto X with the norm not ∈ K P 1 greater than 2kBP . Then AP A X1 P . Since and k are arbitrary, we deduce the proper -deficiency of A fromk thek ≤ right. k | kk k ≤ P Now suppose that A X is normally solvable and A has a finite dimensional kernel contained | 0 in X0 but A is not normally solvable. Let X2 denote a complement of ker A in X. Then the operator A X2 : X2 X is still not normally solvable, but the compression A X3 : X3 X0 to the space| X := →x X : Q x 0 as n which is a complement| of ker A →in X 3 { ∈ 2 k n k → → ∞} 0 is normally solvable, hence C := inf Ax : x X3, x = 1 > 0. Then for every x X2, x = 1 with dist(x, X ) < 1/4 min {k1,C/kA we∈ havek kAx } C/2. Consequently, there∈ is a k k 3 { k k} k k ≥ δ > 0 such that for every > 0 there is an x X2 with x = 1, lim supn (I Fn)x δ and Ax , and the second step again yields the∈ proper k-deficiencyk from thek right.− k ≥ 5thk stepk ≤. It remains to consider operators A which areP normally solvable and dim ker A < . ∗ ∞ Notice that dim coker A = dim ker A in this case, and due to the above considerations A X0 is ∗ | normally solvable, too, hence dim coker A X = dim ker(A X ) . | 0 | 0 Obviously, is a uniform approximate identity on X0, which tends strongly to the identity, P ∗ ∗ ∗ and the sequence = (Pn ) still forms a uniform approximate identity on (X0) . Indeed, for a functional f (XP)∗ and for > 0 let x X , x = 1 be such that f(x) f . Then ∈ 0 ∈ 0 k k | | ≥ k k − P ∗f (P ∗f)(x) = f(P x) f(x) f(Q x) f f Q x , k n k ≥ | n | | n | ≥ | | − | n | ≥ k k − − k kk n k ∗ hence supn Pn f f . k k ≥ k k ∗ Thus, we can apply the previous steps to (A X0 ) and find, for every k dim coker A X0 , a ∗ | ∗ ≤ | -compact rank-k-projection T such that (A X0 ) T = 0. As in the proof of Proposition 1.32 P 0 | 0 ˜0 this yields a -compact projection P of rank k such that P A X0 = 0. For its lifting P which is given by PropositionP 1.33 we find that also P˜0A = 0 by the estimate|

0 0 0 0 0 P˜ A P˜ AP + P˜ AQ P A X P + P˜ AQ , (1.8) k k ≤ k nk k nk ≤ k | 0 kk nk k nk where the ﬁrst term equals zero and the second one tends to zero as n . Thus, we conclude → ∞ that if dim coker A X = then A X and A are properly -deﬁcient from the left. Moreover, | 0 ∞ | 0 P if dim coker A X < then A X is properly -Fredholm, and if, additionally, | 0 ∞ | 0 P

dim coker A = dim coker A X (1.9) | 0 would be true then A would be properly -Fredholm as well. 6th step. If ∗ is an approximate identityP then the idea of the 5th step can be directly applied P to A instead of A X , and the assertion of the theorem is proved for this case. | 0 7th step. If there is a predual space Y and operators (Rn) as supposed in the second condition of the theorem then notice that (Rn) is a uniform approximate identity on Y. Indeed, assume that there is a y Y with supn Rny < y . Then y / Y0. Thus, a bounded linear functional ∈ k k k k ∈ ∗ x X exists with x(y) > 0 and x(Y0) = 0 , that is x = 0 but Pnx = x Rn = 0 for all n, a contradiction.∈ Moreover,| | (R I)R = {P} (P I) ,6 R (R I) =◦ (P I)P , and k n − mk k m n − k k m n − k k n − mk 20 PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

RU = PU for all bounded U R. Thus, the preadjoint operator B has the (Rn)-dichotomy kby whatk k we havek already proved and⊂ the assertion is now also obvious for A. 8th step. We show that the third condition in the theorem implies (1.9). Let dim ker A < as ∞ well as dim coker A X0 < . Then A X0 : X0 X0 is properly -Fredholm and we can choose a | ∞ 0 | → P -regularizer B such that P := I AB (X , ) is a projection parallel to the image of A X . P − ∈ K 0 P | 0 Let B˜ denote the -strong limit of (BPn)n which is in (X, ) (see Theorem 1.19). Further, let P˜0 denote the liftingP of P 0. With the help of Remark 1.18L weP find an m such that I P˜0 AB˜ 2 P (I P˜0 AB˜) k − − k ≤ k m − − k 2 P ((I P 0)P AB˜) + 2 P (I P˜0)Q ≤ k m − n − k k m − nk = 2 P A(BP B˜) + 2 P (I P˜0)Q . k m n − k k m − nk Since both items tend to zero as n we see that P˜0 = I AB˜. Together with P˜0A = 0 due to (1.8) this yields that P˜0 is a projection→ ∞ parallel to im A. Hence− A is Fredholm, together with step 3 also -Fredholm, and the assertion of the theorem is proved as well. 9th step. WeP finally show that the last condition of the theorem implies the third one. For this, let B (X0, ) be properly -Fredholm. Then (BPn)n is bounded and (PkBPn)n is a Cauchy sequence∈ L in (PX, ) for everyP fixed k, since L P P BP P BP P BQ P P k k n − k mk ≤ k k lkk n − mk for n, m l, and P BQ 0 as l , hence there are uniform limits B (X, ) of k k lk → → ∞ k ∈ L P (PkBPn)n∈N for each k. Moreover, due to our assumption, for each x X, there is a uniquely determined -strong limit ∈ P Bx˜ := -lim BPnx in X, Pn→∞ and the mapping x Bx˜ defines a bounded linear operator B˜. Obviously, B˜ X = B and by 7→ | 0 (P B˜ B )x P (B˜ BP )x + P BP B x 0 as n 0, k k − k k ≤ k k − n k k k n − kkk k → → for every x X, we see that P B˜ = B . Hence ∈ k k P (B˜ BP ) = B P BP 0 as n , k k − n k k k − k nk → → ∞ (B˜ BP )P = (B˜ B)P = 0 for n k. k − n kk k − kk Thus, (BP ) converges -strongly to B˜. n P In summary, we see that in the case of uniform approximate identities every operator A (X, ) has the -dichotomy, if X is small (in the sense that ∗ should be an approximate identity)∈ L P or large (inP the sense of being complete w.r.t. -strongP convergence). Until now it is not clear if there really is a gap between these two extremalP cases. The exact formulation of this open question is as follows: Are there a Banach space X, a uniform approximate identity and an operator A (X, ) P ∈ L P which is normally solvable, and fulfills dim ker A < and dim coker A X < dim coker A? ∞ | 0 In the Fredholm theory for band-dominated operators on l∞ the existence of a predual setting 10 played an important role to ensure that the Fredholm properties of A and A X0 coincide. To integrate this aspect into the picture which arises from the present considerations| we note that, by the previous theorem, a predual setting always implies the -dichotomy. On the other hand, P 11 the -dichotomy guarantees the required connection between A and A X : P | 0 10See, for instance, [44] or [15]. 11This particularly clarifies the open problem No. 4 formulated in [15]. 1.3. GEOMETRIC CHARACTERISTICS OF BOUNDED LINEAR OPERATORS 21

Proposition 1.34. Let be a uniform approximate projection on the Banach space X and let P A (X, ) have the -dichotomy. Then A is Fredholm if and only if A X0 is Fredholm. In this∈ case L P P |

dim ker A = dim ker A X , dim coker A = dim coker A X , and ind A = ind A X . | 0 | 0 | 0 Proof. If A is not Fredholm then it is properly -deficient, which also yields the -deficiency P P of A X (X , ) (with regarded as approximate projection on X ) by Proposition 1.33. | 0 ∈ L 0 P P 0 Hence A X0 can not be Fredholm as well, as shown by the fourth condition in Theorem 1.3. Vice versa,| let A be Fredholm and properly -Fredholm. Corollary 1.31 provides an operator B in (X, ) such that ABA = A, BAB = B andP P = I BA, P 0 = I AB (X, ). Then P is a LprojectionP onto the kernel of A and parallel to the range− of B (see Proposition− ∈ K 1.7P). Analogously, P 0 is a projection onto the kernel of B and parallel to the range of A. Thus, ker B is a complement 0 0 of im A, ker A = im P = im P X = ker A X , and ker B = im P = im P X = ker B X . | 0 | 0 | 0 | 0 This proves dim ker A = dim ker A X0 . By Proposition 1.33 we find that the compressions also | 0 fulfill A X B X A X = A X , and the projection P X = I A X B X is onto ker B X and | 0 | 0 | 0 | 0 | 0 − | 0 | 0 | 0 parallel to im A X . Consequently, ker B X is a complement of im A X and we conclude that | 0 | 0 | 0 dim coker A = dim coker A X . The remaining statements now easily follow. | 0 1.3 Geometric characteristics of bounded linear operators 1.3.1 One-sided approximation numbers and their relatives Lower approximation numbers 12 For Banach spaces X, Y and an operator A (X, Y) the r ∈ L m-th approximation number from the right sm(A) and the m-th approximation number from l the left sm(A) of A are defined as sr (A) := inf A F : F (X, Y), dim ker F m , m {k − kL(X,Y) ∈ L ≥ } sl (A) := inf A F : F (X, Y), dim coker F m , m {k − kL(X,Y) ∈ L ≥ } r r r (m = 0, 1, 2, ...), respectively. It is clear that 0 = s0(A) s1(A) s2(A) ... and that the same l ≤ ≤ ≤ holds true for the sequence (sm(A)). Notice that in the case Y = X and dim X < the right- and left-sided variants coincide. ∞

Lower Bernstein and Mityagin numbers Besides the approximation numbers there are further similar geometric characteristics for bounded linear operators A (X, Y) on Banach ∈ L spaces X, Y. Denote by BX the closed unit ball in X and by j(A) := sup τ 0 : Ax τ x for all x X , { ≥ k k ≥ k k ∈ } q(A) := sup τ 0 : A(BX) τBY { ≥ ⊃ } the injection modulus and the surjection modulus, respectively. Due to [54], B.3.8 we have j(A) = inf Ax : x X, x = 1 , j(A∗) = q(A) and q(A∗) = j(A). (1.10) {k k ∈ k k } Furthermore, for given closed subspaces V X and W Y we let J denote the embedding ⊂ ⊂ V map of V into X and by QW the canonical map of Y onto the quotient Y/W . We deﬁne the lower Bernstein and Mityagin numbers by B (A) := sup j(AJ ) : dim X/V < m , m { V } M (A) := sup q(Q A) : dim W < m . m { W } 12The content of Section 1.3 was already published, up to some minor modiﬁcations, in [81]. 22 PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

These characteristics have been discussed in [65], for instance. We will show that there are estimates that connect them to the approximation numbers defined above. Furthermore, these estimates will allow to relate the approximation numbers to the Fredholm property and the norm of the inverse of an operator as well as to the lower singular values in case of X being a Hilbert space. Note also that the sequences (Bm(A)), (Mm(A)) are monotonically increasing and the following auxiliary result holds. Proposition 1.35. Let X, Y be Banach spaces, A (X, Y) and R (Y) be a projection. ∗ ∈ L ∗ ∈ L Then q(Qker RA) = j(A Jim R ). In particular, Mm(A) = Bm(A ) holds for every m N. ∗ ∈ Proof. Define an operator T : im R∗ (Y/ ker R)∗ by T f = g, g(y + ker R) := f(y). T is well defined and surjective since all f im→R∗ vanish on ker R and since every functional g on Y/ ker R ∈ induces a functional g Qker R on Y which, due to Qker R = Qker R R+Qker R (I R) = Qker R R, ◦ ∗ ◦ ◦ − ◦ ∗ even fulfills the relation g Qker R = g Qker R R = R (g Qker R), that is it belongs to im R . Furthermore, T is an isometry◦ since ◦ ◦ ◦ (T f)(Q (y)) f(y) T f = sup | ker R | = sup | | k k y∈Y\ker R Qker R(y) y∈Y\ker R inf y + z k k z∈ker R k k f(y) f(y + z) = sup sup | | = sup sup | | = f , ∈ y + z ∈ y + z k k y∈Y\ker R z ker R k k y∈Y\ker R z ker R k k and (Q∗ (T f)) (y) = (T f)(Q y) = f(y) holds for all f im R∗ and all y Y. Hence ker R ker R ∈ ∈ q(Q A) = j((Q A)∗) = inf A∗Q∗ g : g (Y/ker R)∗, g = 1 ker R ker R {k ker R k ∈ k k } = inf A∗Q∗ (T f) : f im R∗, f = 1 {k ker R k ∈ k k } = inf A∗f : f im R∗, f = 1 = j(A∗J ) {k k ∈ k k } im R∗ follows with (1.10) which gives the first assertion. Now, notice that, by Proposition 1.5, every subspace W Y with dim W < m is the kernel of a certain projection R which provides a subspace U := im⊂R∗ of Y∗. Vice versa, every subspace U Y∗ with dim Y∗/U < m induces a subspace W := y Y : f(y) = 0 f U Y and for⊂ a projection S onto W we see that S∗f = f S = 0 for{ every∈ f U which∀ easily∈ } yields ⊂ that ∗ ◦ ∈ ∗ R := I S fulfills W = ker R and U = im R . Thus, the coincidence of Mm(A) and Bm(A ) follows from− their definitions.

Proposition 1.36. Let X, Y be Banach spaces and A (X, Y). Then, for all m N, ∈ L ∈ sr (A) sl (A) m B (A) sr (A), as well as m M (A) sl (A). 2m 1 ≤ m ≤ m 2m 1 ≤ m ≤ m − − Proof. Let F (X, Y) with dim ker F m and let V X be a subspace with the codimension codim V = dim∈X L/V < m. Then ker F ≥V is at least 1⊂-dimensional. Thus ∩ A F = sup (A F )x : x X, x = 1 sup Ax : x ker F, x = 1 k − k {k − k ∈ k k } ≥ {k k ∈ k k } inf Ax : x (ker F V ), x = 1 inf Ax : x V, x = 1 ≥ {k k ∈ ∩ k k } ≥ {k k ∈ k k } = j(AJV ).

Since V was chosen arbitrarily, it follows that A F Bm(A), and since F is arbitrary, too, we ﬁnd that sr (A) B (A). Now, let > 0.k We− inductivelyk ≥ construct rank-k-projections S m ≥ m k (k = 1, . . . , m) and respective elements x ker S − with x = 1 such that k ∈ k 1 k kk

ck−1 ck := Axk < j(AJker Sk 1 ) + , ≤ k k − 1.3. GEOMETRIC CHARACTERISTICS OF BOUNDED LINEAR OPERATORS 23

∗ as well as functionals f X as follows: Set S := 0 and c := 0. If the projections S ,...,S − k ∈ 0 0 0 k 1 together with their corresponding elements and functionals are already given, we choose xk in ker Sk−1 and a functional f˜k on ker Sk−1 with f˜k = 1 and f˜k(xk) = 1. Furthermore, we deﬁne ˜ k k k the functional fk := fk (I Sk−1) and the projection Skx := i=1 fi(x)xi on X. Notice that ◦ − 13 we obviously have fi(xj) = δij for all i, j k and from ≤ P k−1 f I S − 1 + f k kk ≤ k − k 1k ≤ k ik i=1 X it follows that f 2k−1, hence k kk ≤ k k k i−1 k ASkx = fi(x)Axi ci fi x ck 2 x = ck(2 1) x . k k ≤ k kk k ≤ k k − k k i=1 i=1 i=1 X X X

Defining V := ker Sm −1, we find sr (A) = inf A F : dim ker F m A A(I S ) m {k − k ≥ } ≤ k − − m k (1.11) = AS c (2m 1) < (2m 1)(j(AJ ) + ) (2m 1)(B (A) + ), k mk ≤ m − − V ≤ − m which completes the proof of the first estimate, since is arbitrary. ∗ ∗ r ∗ From Proposition 1.35 and the above applied to A we know that Mm(A) = Bm(A ) sm(A ). ≤l Fix > 0 and an operator F (X, Y) with dim coker F m such that A F sm(A) + . Choose a rank m projection ∈Q Lsuch that QF < (see≥ Proposition 1.6k).− Thenk ≤ the kernel of F ∗(I Q)∗ is at least m-dimensional and k k − sr (A∗) A∗ F ∗(I Q)∗ A∗ F ∗ + F ∗Q∗ = A F + QF sl (A) + 2 m ≤ k − − k ≤ k − k k k k − k k k ≤ m l gives the asserted estimate Mm(A) sm(A) since was chosen arbitrarily. For the remaining part of the second≤ estimate in the assertion we use a construction similar to the one above. Fix > 0 and set R0 := 0, d0 := 0. For k = 1, . . . , m we gradually choose functionals g ker R∗ with g = 1 such that k ∈ k−1 k kk d d := A∗g < j(A∗J ) + , k−1 k k ker Rk∗ 1 ≤ k k − as well as elements y˜k Y with y˜k = 1 and gk(˜yk) 1 , respectively. Furthermore, we −1∈ k k | | ≥ − k define y := (g (˜y )) (I R − )˜y and an operator R y := g (y)y on Y each time. We k k k − k 1 k k i=1 i i easily check that gi(yj) = δij for all i, j k hence Rk is a projection of rank k, and from ≤ P − 1 1 k 1 y I R − 1 + y k kk ≤ 1 k − k 1k ≤ 1 k ik i=1 ! − − X k 1 we conclude y 2 − . Thus, for all g Y∗ and all x X, k kk ≤ (1−)k ∈ ∈ k k ∗ ∗ ∗ (A Rkg)(x) = g(RkAx) = gi(Ax)g(yi) = g(yi)A gi (x) k k k k ! i=1 i=1 X X k k 2i− 1 2k 1 g y d x d g x d − g x . ≤ k kk ik ik k ≤ k (1 )i k kk k ≤ k (1 )k k kk k i=1 i=1 X X − − 13 δij = 1 if i = j, and δij = 0 otherwise 24 PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

The assertion then follows by 2m 1 sl (A) = inf A F : dim coker F m R A = A∗R∗ d − m {k − k ≥ } ≤ k m k k mk ≤ m (1 )m − 2m 1 2m 1 − (B (A∗) + ) = − (M (A) + ) ≤ (1 )m m (1 )m m − − where > 0 is arbitrary.

r l ∗ Now we have s1(A) = B1(A) = j(A) and s1(A) = M1(A) = q(A) = j(A ), hence we deduce Corollary 1.37. Let A (X, Y). Then ∈ L −1 −1 r/l A if A is invertible s1 (A) = k k (0 if A is not invertible from the left/right. With the help of Proposition 1.6 we ﬁnd even more.

Corollary 1.38. Let A (X, Y). ∈ L 1. If A is normally solvable and k is greater than the dimension of the kernel (cokernel) of A r l r l then sk(A) (sk(A), resp.) is non-zero. Otherwise sk(A) (sk(A)) is equal to zero. 2. If A is not normally solvable then all approximation numbers are equal to zero. In particular, A is Fredholm if and only if the number of vanishing approximation numbers of A is ﬁnite.

1.3.2 Hilbert spaces and lower singular values Suppose now that X, Y are Hilbert spaces and A (X, Y). We denote by A? the Hilbert adjoint as introduced in Section 1.1.1 and consider the essential∈ L spectrum of the operator A?A which is 14 ? a subset of R+, the set of all non-negative real numbers. Furthermore, let d := inf σess(A A) and p σ (A) σ (A) ... 1 ≤ 2 ≤ be the sequence of the non-negative square roots of the eigenvalues of A?A less than d, counted according to their multiplicities. If there are only N A (= 0, 1, 2,...) such eigenvalues, we put σN A+1(A) = σN A+2(A) = ... = d. These numbers may be called the lower singular values of A.

Corollary 1.39. Let X, Y be Hilbert spaces and A (X, Y). Then, for all m, ∈ L r l ? sm(A) = Bm(A) = σm(A), as well as sm(A) = Mm(A) = σm(A ). Proof. Firstly, suppose that A is normally solvable and has a trivial kernel. Set B := A?A and for a subspace U X let U ⊥ denote its orthogonal complement. Then ⊂ (B (A))2 = sup inf Ax 2 : x U ⊥, x = 1 : dim U < m m { {k k ∈ k k } } = sup inf x, Bx : x U ⊥, x = 1 : dim U < m { {h i ∈ k k } } 2 which equals (σm(A)) by the Min-Max Principle (see [66], Theorem XIII.1 or below).

14See the deﬁnition in Section 1.4 and Theorem 1.40, and note that (A?A)? = A?A. 1.3. GEOMETRIC CHARACTERISTICS OF BOUNDED LINEAR OPERATORS 25

If A is not normally solvable or dim ker A = then sr (A) = B (A) = 0 for all m, due to ∞ m m Corollary 1.38. Moreover, B is not Fredholm in this case, that is 0 σess(B) hence σm(A) = 0 for all m. ∈ In the case that is normally solvable and we set ˆ and find A k = dim ker A < A := AJ(ker A)⊥ that ∞ Bm+k(A) = Bm(Aˆ) = σm(Aˆ) = σm+k(A) for all m, as well as B (A) = σ (A) = 0 for all l k. l l ≤ Moreover, for a subspace U of dimension m and PU the orthogonal projection onto U, (sr (A))2 AP 2 = sup Ax, Ax : x U, x 1 m ≤ k U k {h i ∈ k k ≤ } (1.12) sup x Bx : x U, x 1 BJ . ≤ {k kk k ∈ k k ≤ } ≤ k U k A m 2 If m N , then there is an orthonormal system xi i=1 of eigenvectors, i.e. Bxi = (σi(A)) xi. Set U≤:= span x m and deduce from (1.12) that{ } { i}i=1 m m (sr (A))2 sup α2(σ (A))2 : x = α x , x 1 (σ (A))2. m ≤ i i k k k k ≤ ≤ m (i=1 ) X kX=1 A 2 2 For the case m > N we note that C := B d I is not Fredholm, since d σess(B). That is, C is not normally solvable or dim ker C = dim coker− C = . Thus, by Proposition∈ 1.6, for every > 0 ∞ r 2 2 there is a subspace U of dimension m such that CJU < , hence (sm(A)) BJU < d + . Together with Proposition 1.36 this finishes thek proofk of the first assertion, and≤ k the secondk one l r ? follows by duality, Proposition 1.35 and since sm(A) = sm(A ) in the case of Hilbert spaces. For completeness we state the Min-Max-Principle from [66], Theorem XIII.1 here, where we call H (X) self-adjoint, if H = H?, that means Hx, y = x, Hy for all x, y X. ∈ L h i h i ∈ Theorem 1.40. Let H (X) be a normally solvable, injective, and self-adjoint operator on ∈ L ⊥ the Hilbert space X and define µn(H) := supU inf x, Hx : x U , x = 1 , the supremum over all subspaces U of dimension at most n 1.{h Then,i for each∈ fixedk kn, exactly} one of the following hold: − There are n eigenvalues (counted according to their algebraic multiplicities) below the bottom • of the essential spectrum, and µn(H) is the nth eigenvalue. µ (H) is the bottom of the essential spectrum. • n Remark 1.41. Besides the lower approximation, Bernstein and Mityagin numbers there are also their much more famous “upper relatives”. For example, the upper approximation numbers sk(A) of an operator A are given by s (A) := inf A F : F (X, Y), rank F < k . k {k − kL(X,Y) ∈ L } These numbers form a decreasing sequence A = s0(A) s1(A) ... 0 and, roughly speaking, they sound out the spectrum of an operatork k from above,≥ and not≥ from≥ below as the lower approximation numbers do. For the other characteristics the definitions are similar. Notice that all of these “big brothers” are so-called s-numbers, for which a well developed theory is available that includes many results on the relations between them. The modern books [53] and [22] may provide a good overview and introduction to that business. Unfortunately, as far as we know, there are no results in the literature for the lower versions in the case of (infinite dimensional) Banach spaces, except those of [65], from where we borrowed the equalities Bm(A) = σm(A) and ? Mm(A) = σm(A ). In particular, note that the estimates in Proposition 1.36 seem hardly to be optimal. Nevertheless, they suffice for the aims of the present text. 26 PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

1.4 Spectral properties 1.4.1 The spectrum Let be a Banach algebra with identity I and A . The spectrum sp A of A is the set of A ∈ A all complex numbers z C such that A zI is not invertible in . From any textbook on functional analysis or spectral∈ theory it is well− known that the spectrumA is a non-empty compact set. Hence, the spectral radius ρ(A) of A ρ(A) := sup z : z sp A {| | ∈ } is well defined and the Spectral Radius Formula states that ρ(A) = inf An 1/n = lim An 1/n. n∈N k k n→∞ k k We particularly get the spectrum of a bounded linear operator A (X) on a Banach space X: ∈ L sp A := z C : A zI is not invertible . { ∈ − } Further, the essential spectrum spess A of A is defined as the spectrum of A+ (X) in the Calkin algebra (X)/ (X) or, equivalently due to Theorem 1.3, K L K spess A := z C : A zI is not Fredholm . { ∈ − } 1.4.2 The (N, )-pseudospectrum In [30] the authors write “A computer working with finite accuracy cannot distinguish between a non-invertible matrix and an invertible matrix the inverse of which has a very large norm”. Therefore one replaces the spectrum by so-called pseudospectra which reflect finite accuracy. For some pioneering work we refer to Landau [40], [41], Reichel, Trefethen [67] and Böttcher [6], and for a (or more striking: the) comprehensive source we recommend the monograph [87] of Trefethen and Embree.

Deﬁnition 1.42. For N Z+ and > 0 the (N, )-pseudospectrum of a bounded linear operator A on a Banach space X is∈ deﬁned as the set

N N −2 2− 15 spN, A := z C : (A zI) 1/ . { ∈ k − k ≥ } Remark 1.43. Notice that (for N = 0) this definition of the (N, )-pseudospectrum includes the definition of the (classical) -pseudospectrum −1 sp A := z C : (A zI) 1/ . { ∈ k − k ≥ } The -pseudospectra have gained attention after [67] and [6] disclosed that, on the one hand they approximate the spectrum but are less sensitive to perturbations, and on the other hand the -pseudospectra of discrete convolution operators mimic exactly the -pseudospectrum of an appropriate limiting operator, which is in general not true for the “usual” spectrum. See also [4], [13], [30], [11] and the references cited there. Later on, Hansen [32], [33] introduced the (N, )-pseudospectra for linear operators on separable Hilbert spaces and pointed out that they share several nice properties with case N = 0, but offer a better insight into the approximation of the spectrum. Furthermore, it was shown how the spectrum can be approximated numerically, based on the consideration of singular values of certain finite matrices. We now recover and extend these results in what follows and in Section 3.2.3. 15Here we use the convention kB−1k = ∞ if B is not invertible. 1.4. SPECTRAL PROPERTIES 27

Within this section we want to show that the (N, )-pseudospectra provide a nice tool for the approximation of the spectrum of a bounded linear operator A, even in the Banach space case. More precisely, we will show that Theorem 1.44. Let A (X). For every δ > > 0 there is an N such that, for all N N , ∈ L 0 ≥ 0 B (sp A) sp A B (sp A), (1.13) ⊂ N, ⊂ δ where B(S) := z C : dist(z, S) denotes the closed -neighborhood of the set S. { ∈ ≤ } In a very recent preprint [34] Hansen and Nevanlinna mentioned that this result is in force even in the Banach space context, but the Hilbert space approach for the approximate determination of the spectrum via singular values of ﬁnite matrices cannot be extended to the Banach case since there is no involution available anymore. Therefore we propose a modiﬁcation of Hansens idea which replaces the singular values by the injection and surjection modulus. Here comes the precise description:

If there is a sequence (Ln) of compact projections which converge -strongly to the identity then we will obtain an approximation of the spectrum of A based on much∗ simpler operators: For this deﬁne the functions

N N N 2− γm,n(z) := min j((L (A zI)L )2 J ), q(Q (L (A zI)L )2 ) (1.14) N { n − n im Lm ker Lm n − n } where the operators J, Q are deﬁned as in Section 1.3.1. We are interested in their sublevel sets

m,n m,n Γ := z C : γ (z) N, { ∈ N ≤ } because for every fixed β > > α > 0 and sufficiently large N, m, n we will get B (sp A) Γm,n B (sp A). (1.15) α ⊂ N, ⊂ β Notice that the projections Ln are of finite rank, thus the operators which have to be considered m,n for the evaluation of γN (z) are operators acting on finite dimensional spaces. For the Hilbert space case this particularly means that we only have to consider the smallest singular values of m,n certain finite matrices in order to determine ΓN, (by Corollary 1.39). The paper [31] contains an extensive discussion of this case including explicit algorithms and examples. So, we omit a detailed repetition here and focus on the theoretical treatment of the Banach space case instead. Remark 1.45. Actually, we prove the latter for the case that X possesses a uniform approximate identity with BP = 1 such that X has the -dichotomy, and the sequence (Ln) of -compact projectionsP converges -strongly to the identity.P In such a setting the asserted relationP (1.15) is true for A (X, P). Anyway, the proofs work in both cases. In [78] the author already published these∈ results,L P proofs and discussions of Section 1.4 for the “classical” setting without and under the additional condition L = 1 for all n. P k nk A level function for the (N, )-pseudospectrum Define

N N (A zI)−2 −2− if z / sp A γN (z) := k − k ∈ , and γ(z) := dist(z, sp A). 0 if z sp A ( ∈ The function γ is continuous everywhere, equals zero in all points z sp A, and ∈ 1 1 γ(z) = − = N N = lim γN (z) ρ((A zI) 1) lim (A zI)−2 2− N→∞ − N→∞ k − k 28 PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

for every z / sp A. The last equation is the deﬁnition of γN (z), the second one is given by the Spectral Radius∈ Formula, and for the ﬁrst one consider

1 (A zI)−1 yI = (A zI)−1 [I y(A zI)] = (A zI)−1y z + I A . − − − − − − y − We see that (A zI)−1 yI is not invertible if and only if z + 1 belongs to the spectrum of A, − − y that is sp((A zI)−1) = 1 : x sp A , and thus − x−z ∈ n o 1 1 1 ρ((A zI)−1) = sup : x sp A = = . − x z ∈ inf x z : x sp A dist(z, sp A) − {| − | ∈ }

The γ are continuous in every z / sp A and (pointwise) monotonically increasing w.r.t. N since N ∈

N+1 (N+1) N (N+1) N N (A zI)−2 2− ( (A zI)−2 2)2− = (A zI)−2 2− . (1.16) k − k ≤ k − k k − k Combining these observations we see that 0 γ (z) γ(z), the functions γ are continuous ≤ N ≤ N everywhere, and γN (z) converges increasingly to γ(z) for every z C. By Dini’s Theorem this ∈ even gives uniform convergence on every compact subset of C. Fix δ > > 0. It is clear from the deﬁnition that z sp A if and only if γ (z) . Choose ∈ N, N ≤ r > 0 large enough to guarantee that γN (z) > for all z C Ur(0) and all N. This is possible by a Neumann series argument since A is bounded. Then∈ we have\ uniform increasing convergence of γN (z) to γ(z) on Br(0). Thus, there is an N0 such that for every N N0 and every z C ≥ ∈ γ(z) γ (z) γ(z) δ, ≤ ⇒ N ≤ ⇒ ≤ which yields (1.13) and ﬁnishes the proof of Theorem 1.44.

Uniform approximations for γN (z) Notice that by Corollary 1.37

N N N 2− γ (z) = min j((A zI)2 ), q((A zI)2 ) . N { − − } We now use this representation as a starting point for the definition of approximating substitutes: Let be a uniform approximate identity on the Banach space X with BP = 1 and suppose P that X has the -dichotomy. Further suppose that (Ln) is a sequence of -compact projections which converge P -strongly to the identity. For A (X, ) set P P ∈ L P N N N 2− γm(z) := min j((A zI)2 J ), q(Q (A zI)2 ) . N { − im Lm ker Lm − } m Proposition 1.46. For every N and m, the functions γN (z) are continuous w.r.t. z C. m ∈ Further, for every fixed point z C, the sequence (γ (z))m is bounded below by γN (z) and ∈ N converges to γN (z). The convergence is even uniform on every compact subset of C. We first state an auxiliary result.

Proposition 1.47. Let X, Y, Z be Banach spaces and A (X, Y) as well as B (Y, Z). Then j(BA) j(B) j(A) and q(BA) q(B) q(A). ∈ L ∈ L ≥ ≥ 1.4. SPECTRAL PROPERTIES 29

Proof. If one of the numbers j(A), j(B), q(A), q(B) equals zero then the respective assertion is obviously true. So, let these four numbers be strictly positive and check that

BAx j(BA) = inf BAx : x X, x = 1 = inf k k Ax : x X, x = 1 {k k ∈ k k } Ax k k ∈ k k k k inf By : y Y, y = 1 inf Ax : x X, x = 1 = j(B) j(A). ≥ {k k ∈ k k } {k k ∈ k k }

Further, ﬁx σ < q(B) and τ < q(A). Then σBZ B(BY) and τBY A(BX). Consequently, ⊂ ⊂

στBZ τB(BY) = B(τBY) B(A(BX)) = BA(BX) ⊂ ⊂ which shows that στ q(BA) and ﬁnishes the proof. ≤ Proof of Proposition 1.46. The continuity is obvious by the relations (1.10), and we immediately conclude the estimate j(BJim Lm ) j(B)j(Jim Lm ) = j(B) for every m from the previous result. With Proposition 1.35 also ≥ holds. These q(Qker Lm B) q(Qker Lm )q(B) = j(Jim Lm∗ )q(B) = q(B) ≥ N observations particularly hold for all B := (A zI)2 and hence, in every z, we get the lower m − bound γN (z) for (γN (z))m. If B (X, ) is invertible then, due to Theorem 1.14, B−1 belongs to (X, ) and we have ∈ L P −1 −1 −1 −1 L P j(B) = q(B) = B = B X0 by Proposition 1.33, since BP = 1. Given δ > 0 choose k k k −| 1 k −1 −1 −1 −1 −1 x X0, x = 1, such that B X0 > B x δ and set y := B x, w := y y. Then∈ k k k | k k k − k k

−1 −1 x By j(B) = q(B) = B X > k k δ = k k δ = Bw δ. k | 0 k B−1x − y − k k − k k k k The latter can be further estimated by

Bw BL w B(I L )w k k ≥ k m k − k − m k BL w 1 L w = k m k − k m k BL w B(I L )w L w − L w k m k − k − m k (1.17) k m k k m k (I L )w j(BJ ) k − m k BL B(I L )w ≥ im Lm − 1 (I L )w k mk − k − m k − k − m k where (I L )w 0 as m since w X and (L ) converges -strongly to the identity. k − m k → → ∞ ∈ 0 m P Also note that (Lm) is bounded, due to Theorem 1.19. Since δ was chosen arbitrarily, we ﬁnd that j(BJ ) j(B) as m . Plugging this in the estimate im Lm → → ∞ N N N 2− N 2− j((A zI)2 J ) γm(z) γ (z) = j((A zI)2 ) , z / sp A − im Lm ≥ N ≥ N − ∈ we deduce that γm(z) converges to γ (z) for every z / sp A. N N ∈ Now, let B be -deﬁcient from the right, or let the kernel of B be non-trivial. Fix δ > 0 and P a -compact rank-one-projection P such that BP < δ. Let w im P X0, w = 1 and applyP (1.17) to deduce k k ∈ ⊂ k k

(I L )w δ > Bw j(BJ ) k − m k BL B(I L )w , k k ≥ im Lm − 1 (I L )w k mk − k − m k − k − m k which yields j(BJ ) 0 as m also in this case, since δ > 0 was chosen arbitrarily. im Lm → → ∞ 30 PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

If B is -deﬁcient from the left, or the cokernel of B is non-trivial then, for given δ > 0, we choose aP -compact rank-one-projection P such that PB < δ as well as a functional f im P ∗, f = 1,P and ﬁnd again by (1.17) and Proposition 1.35k thatk ∈ k k ∗ ∗ ∗ (I Lm)f ∗ ∗ ∗ ∗ δ > B f j(B Jim L ) k − k B L B (I L )f k k ≥ m∗ − 1 (I L∗ )f k mk − k − m k − k − m k (I L∗ )P ∗ q(Q B) k − m k B∗L∗ B∗ (I L∗ )P ∗ . ≥ ker Lm − 1 (I L∗ )P ∗ k mk − k kk − m k − k − m k

The -strong convergence of (Lm) together with the -compactness of P prove q(Qker Lm B) 0 as mP in that case, too. Since X has the -dichotomy,P there are no further possible cases,→ → ∞ P m hence we get pointwise convergence of the continuous functions γN (z) to the continuous function γN (z) in the whole complex plane. m Let M C be a compact set. It remains to show the uniform convergence of γN to γN on ⊂ m M. For this we construct a sequence (fN )m of continuous functions deﬁned on M such that m m m fN (z) γN (z) γN (z) and fN (z) converge decreasingly to γN (z) for every z M. Then ≥ ≥ m ∈ Dini’s Theorem applies to (fN ), provides its uniform convergence, and hence also the uniform m m convergence of (γN ). So, let us construct the desired functions (fN ). Let k N. For each 2N ∈ B := (A zI) : z M there is a w im Lk, w = 1 such that, using the estimate (1.17) again,∈ R { − ∈ } ∈ k k 1 j(BJ ) = inf Bx : x im L , x = 1 Bw im Lk {k k ∈ k k k } ≥ k k − 2k (I L )w 1 j(BJ ) k − m k BL B(I L )w ≥ im Lm − 1 (I L )w k mk − k − m k − 2k − k − m k (I L )L 1 j(BJ ) k − m kk BL B (I L )L . ≥ im Lm − 1 (I L )L k mk − k kk − m kk − 2k − k − m kk Recall that the operator L is -compact and the sequence (L ) converges -strongly to the k P m P identity. Hence, there is an mk N such that, for every operator B in the bounded set and ∈ R every m mk, the estimate j(BJim LK ) j(BJim Lm ) 1/k holds. Without loss of generality we can choose≥ these m such that m 2≥k for every k,− and we conclude k k ≥ 2 2 j(BJim Lk ) + j(BJim Lm ) + for all k N, m mk and B . k ≥ m ∈ ≥ ∈ R The same arguments applied to j(B∗J ), with Proposition 1.35, yield numbers m˜ s.t. im Lk∗ k 2 2 q(Qker Lk B) + q(Qker Lm B) + for all k N, m m˜ k and B . k ≥ m ∈ ≥ ∈ R

Consequently, there is a strictly increasing sequence (ln) N such that ⊂ 2 2 min j(BJim Lln ), q(Qker Lln B) + min j(BJim Ls ), q(Qker Ls B) + { } ln ≥ { } s m for all n N, B and s ln+1. For ln+1 m < ln+2 we now deﬁne the functions f by ∈ ∈ R ≥ ≤ N N 2− m 2N 2N 2 f (z) := min j((A zI) Jim L ), q(Qker L (A zI) ) + N { − ln ln − } l n and straightforwardly check that they meet all requirements: They can be written in the form N N f m = ((γln )2 + 2 )2− , hence they are continuous on M and converge pointwise to γ by what N N ln N 1.4. SPECTRAL PROPERTIES 31

we have already proved. Due to the special choice of the (ln), this convergence is monotonically decreasing and the functions even fulﬁll f m(z) γm(z) on M. N ≥ N Until now, we have seen that the function γ can be approximated in a sense by the functions γN , m and further the functions γN are approximations of γN . As a third step we ﬁnally approximate m m,n γN by the functions γN given in (1.14):

N N N 2− γm,n(z) = min j((L (A zI)L )2 J ), q(Q (L (A zI)L )2 ) . N { n − n im Lm ker Lm n − n } 2N For A (X, ) it is clear that the sequence ((Ln(A zI)Ln) )n∈N converges -strongly, hence the∈ L argumentsP of j( ) and q( ) converge in the norm− as n for every z sinceP L is · · → ∞ m -compact and Qker Lm = Qker Lm Lm holds. This convergence is even uniform with respect toP z on every compact set, since these◦ arguments are polynomials in z whose (operator-valued) coeﬃcients converge in the norm. This provides the uniform convergence γm,n(z) γm(z) as N → N n on every compact subset of C. → ∞ Regard the compact set M := B2kAk+4β(0). For given β > > α > 0 we choose N0 such that γ(z) γN (z) > (β )/2 for all N N0 and z M. Furthermore, for N N0, we choose m (N−) such that γm−(z) γ (z) < (≥ α)/2 for∈ all z M and all m m ≥(N). Finally, we 0 N − N − ∈ ≥ 0 take n0(N, m) large enough to guarantee that for all n n0(N, m) and z M the estimate γm,n(z) γm(z) < min ( α), (β ) /2 and, additionally,≥ j(L J ), q(∈Q L ) 1/2 | N − N | { − − } n im Lm ker Lm n ≥ hold. The latter is again possible since (Ln) converges -strongly to the identity and Lm is -compact. Then, for every z M, P P ∈ α α γ(z) α γ (z) α γm(z) α + − γm,n(z) α + 2 − = ≤ ⇒ N ≤ ⇒ N ≤ 2 ⇒ N ≤ 2 and β β β γm,n(z) γm(z) + − γ (z) + − γ(z) + 2 − = β. N ≤ ⇒ N ≤ 2 ⇒ N ≤ 2 ⇒ ≤ 2 Applying Proposition 1.47 we get for arbitrary bounded linear operators B that

j(BL J ) = j((BL )(L J )) j(BJ )j(L J ) j(BJ )/2 n im Lm n n im Lm ≥ im Ln n im Lm ≥ im Ln and analogously q(Qker Lm LnB) q(Qker Ln B)/2. Thus, it is immediate from the definitions and an estimate similar to (1.16) that,≥ for z / M, ∈ m,n n,n n,n −1 −1 −1 2γ (z) γ (z) γ (z) = z (Ln z LnALn) > 2β. N ≥ N ≥ 0 | |k − kL(im Ln) The estimate γ(z) > β for z / M is also obvious. Consequently, for suitably chosen N, m and n, we arrive at (1.15). Let us condense∈ this outcome in a theorem. Theorem 1.48. Let be a uniform approximate identity on the Banach space X providing X P with the -dichotomy, BP = 1, (Ln) be a sequence of -compact projections tending -strongly to the identity,P and let A (X, ). For every fixedPβ > > α > 0 and all sufficientlyP large N N , m m (N) 16 and∈ Ln nP (N, m) the set ≥ 0 ≥ 0 ≥ 0 m,n 2N 2N 2N Γ = z C : min j((Ln(A zI)Ln) Jim Lm ), q(Qker Lm (Ln(A zI)Ln) ) N, ∈ { − − } ≤ n o fulfills B (sp A) Γm,n B (sp A). α ⊂ N, ⊂ β 16 m0(N) means that m0 depends on N. 32 PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

Notice that, if X is a separable Hilbert space and the Ln are compact, orthogonal projections which are nested in the sense that LnLn+1 = Ln+1Ln = Ln for all n N and converge strongly 17 ∈ to the identity, then = (Ln) satisﬁes the conditions of this section. Corollary 1.39 further yields P

m,n 2N ? 2N 2N Γ = z C : min σ1((Ln(A zI)Ln) Jim Lm ), σ1((Ln(A zI¯ )Ln) Jim Lm ) . N, ∈ { − − } ≤ n o We again note that the classical Hilbert space setting and the idea of approximating the spectrum via the consideration of certain “rectangular matrices” and their smallest singular value is subject of [31].

1.4.3 Convergence of sets

Definition 1.49. The Hausdorff distance of two compact sets S, T C is defined as ⊂

dH (S, T ) := max max dist(s, T ), max dist(t, S) . s∈S t∈T

This function dH forms actually a metric on the set of all non-empty compact subsets of C, and oﬀers an alternative view on the present results. Remember that the sets sp A, B(sp A), spN, A n,m and ΓN, are compact. Corollary 1.50. Let A (X). Then ∈ L

lim dH (sp A, B(sp A)) = 0 and lim dH (B(sp A), spN,(A)) = 0. →0 N→∞

Thus, the (N, )-pseudospectra converge to the spectrum of A with respect to the Hausdorﬀ distance if N and 0. Moreover, → ∞ → m,n lim lim sup lim sup dH (B(sp A), ΓN, ) = 0, N→∞ m→∞ n→∞

m,n that is, the spectrum of A can be approximated w.r.t. the Hausdorﬀ metric even by the sets ΓN, . Proof. Firstly, note that for compact sets S, T, U with S T U we have ⊂ ⊂

dH (S, T ) = max dist(t, S) max dist(u, S) = dH (S, U). t∈T ≤ u∈U

Clearly, dH (sp A, B(sp A)) which proves the ﬁrst assertion. For the second one apply Theorem 1.44 and the estimate≤

d (B (sp A), sp (A)) d (B (sp A),B (sp A)) δ , H N, ≤ H δ ≤ − which holds for every ﬁxed δ > > 0 and suﬃciently large N. For the last assertion we employ Theorem 1.48 in the same way.

Remark 1.51. We want to point out that the -pseudospectrum does not behave continuously with respect to the parameter in general, that is the inclusion

−1 clos z C : (A zI) > 1/ sp A { ∈ k − k } ⊂ 17 The sequence (Ln) is called a filtration in that case. 1.4. SPECTRAL PROPERTIES 33 can be proper. Shargorodsky addressed a paper [84] to this fact, in which an explanation and appropriate examples are given in great detail. The point is that (A zI)−1 , the resolvent norm, can take constant values on open sets. Actually, the historyk of the− investigationk of this phenomenon is much longer. Globevnik [26] posed the question “Can (λe a)−1 be constant on an open subset of the resolvent set” for an element a of a complex Banachk − algebra.k He could only derive a partial answer, but he was able to show that the answer is No for the resolvent norm of a bounded linear operator on a complex uniformly convex Banach space.18 Unfortunately, it remained rather unnoticed, and so this question emerged again in the 90’s where, independently from the earlier, Böttcher asked this and, together with Daniluk, he tackled the case of bounded linear operators on Hilbert spaces in [6], Proposition 6.1 (see also [18]), and later on the Lp-spaces with 1 < p < as well. Shargorodsky combined all these observations and completed the picture in [84], where∞ he pointed out that Hilbert spaces and the Lp-spaces are complex uniformly convex by [17], and that the outcome even extends to Banach spaces which have a complex uniformly convex dual space. This particularly permits to cover all Lp-spaces with p [1, ]. ∈ ∞ Of course, the phenomenon of jumping pseudospectra (if it exists in the underlying setting) is reflected in the behavior of the (N, )-pseudospectra as well but, as Theorem 1.44 and Corollary 1.50 show, it gets less significant in any case, since the difference of the respective sets

N N −2 2− clos z C : (A zI) > 1/ spN, A { ∈ k − k } ⊂ becomes small with growing N.

1.4.4 Quasi-diagonal operators The above observation that we can approximate the spectrum with the help of “rectangular finite sections” raises hope that this may even be possible by the usual finite sections. In fact, this is not true as a simple and well known example demonstrates: Let V denote the shift operator p (xi)i∈Z (xi+1)i∈Z on l . It is invertible, has norm 1 and its inverse has norm 1 as well. Hence V αI 7→is invertible whenever α = 1 by a Neumann series argument. On the other hand, the spectrum− of every finite section| equals| 6 0 and also their (N, )-pseudospectra contain 0, whereas the (N, )-pseudospectra of V approximate{ } the unit circle. Actually, there is an explanation and a solution to this dilemma, and we will take up this matter again, after we have the required tools of the Parts2 and3 available. For the moment we turn our attention to a class of operators which accomplish our desire.

Deﬁnition 1.52. Let (Ln) be a sequence of -compact projections tending -strongly to the identity, where = (P ) is a uniform approximateP identity on X, which equipsP X with the P n -dichotomy and fulﬁlls BP = 1. An operator A (X, ) is called quasi-diagonal with respect P ∈ L P to (Ln) (or (Ln) is said to quasi-diagonalize A) if

[A, L ] := AL L A 0 as n . k n k k n − n k → → ∞ Note that by a result of Berg [3] every normal operator A on a separable Hilbert space, hence also every self-adjoint operator, has a sequence of orthogonal projections which quasi-diagonalizes A. We will pay some more attention to such operators and discuss further details in Section 4.1. At this stage we restrict our considerations solely to their pseudospectra.

18For a deﬁnition see Section 3.2.3 where we will have a closer look at this notion. 34 PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

Theorem 1.53. Let A be quasi-diagonal with respect to (Ln). Then

0 = lim dH (sp A, B(sp A)) →0

= lim lim dH (sp A, spN, A) →0 N→∞

= lim lim lim sup dH (sp A, spN,(LmALm)). →0 N→∞ m→∞

Proof. The ﬁrst and second equation are done by Corollary 1.50. For the last one note that Γm,m = sp (L AL ) and that γm γm,m already converges uniformly to zero on every N, N, m m N − N compact subset of C as m since A is quasi-diagonal. → ∞ We mention that Brown [14] already proved the convergence of the classical -pseudospectra in the Hilbert case, based on the results of the C∗-algebra approach of Hagen, Roch and Silbermann

[30]. For those we have the relation >0 sp A = sp A to the usual spectrum. Also Hansen [32] considered the spectral approximation of self-adjoint operators, taking their quasi-diagonality as well as their (N, )-pseudospectra intoT account.

1.5 Example: operators on lp-spaces

Let us summarize what we have already found for the spaces lp = lp(Z,X) and the sequence = (P ) of projections given by P n

P :(x ) (..., 0, x− , . . . , x , 0,...). m i 7→ m m For every 1 p and every Banach space X, forms a uniform approximate identity and lp has the ≤-dichotomy≤ ∞ (cf. Section 1.2.3 and CorollaryP 1.28). Pp Let A (l , ) and suppose that the finite section sequence (PnAPn) is stable. Then, for every b lp,∈ the L equationP Ax = b has a unique solution x lp, and also the equations P AP x = P b ∈ ∈ n n n n have unique solutions xn (for sufficiently large n), which converge -strongly to x (Corollary 1.23 and Proposition 1.29). P So, all we are left with is the question how to prove the stability of the finite section sequence. We want to discuss this for a more specific class of operators.

1.5.1 Band-dominated operators

∞ p Every sequence a = (ai) l (Z, (X)) gives rise to an operator aI (l ) (a so-called multiplication operator) via ∈ L ∈ L (ax)i = aixi, i Z. ∈ ∞ Evidently, aI L(lp) = a ∞. By this means, the sequences in l (Z, C) induce multiplication operators ask well.k k k

Deﬁnition 1.54. A band operator A is a ﬁnite sum of the form A = aαVα, where α Z, α ∈ aα are multiplication operators, and Vα denote the shift operators P p (Vαf)(x) := f(x α), x Z, f l . − ∈ ∈ An operator is called band-dominated if it is the uniform limit of a sequence of band operators. We denote the class of all band-dominated operators by p . Al 1.5. EXAMPLE: OPERATORS ON LP -SPACES 35

Clearly, in case X = C, the matrix representation of a multiplication operator aI with respect p to the canonical basis in l is an infinite diagonal matrix with the numbers ai along the main diagonal. The matrix representations of band operators are matrices having a finite bandwidth, that is there are only finitely many non-zero (sub- or super-)diagonals. This is why we call the uniform limits in p band-dominated. To state a collection of important properties of Al p and its elements we let BUC denote the algebra of all bounded and uniformly continuous Al functions on the real line. For ϕ BUC, t > 0 and r R we define with ϕt(x) := ϕ(tx) and ∈ ∈ ϕt,r(x) := ϕt(x r) certain inflated and shifted versions of ϕ. Furthermore, let ϕˆ stand for − ∞ the sequence (ϕ(n))n∈Z which obviously belongs to l , and therefore induces a multiplication operator ϕIˆ . The following results are well known, and can be found in [63], Theorem 2.1.6 and Propositions 2.1.7ff, and in [44], Theorem 1.42. Theorem 1.55. . p p 1. The set p forms a closed and inverse closed subalgebra of (l , ) and := (l , ), Al L P K K P the set of all -compact operators, is a closed ideal in lp . Furthermore, lp / is inverse closed in theP quotient algebra (lp, )/ . A A K L P K 2. For an operator A (lp) the following are equivalent: ∈ L A is band-dominated. • For every > 0, there exists an M > 0 such that whenever F,G are subsets of Z with • dist(F,G) := inf x y : x F, y G > M, then χ Aχ I p < . {k − k ∈ ∈ } k F G kL(l ) For every ϕ BUC, • ∈ lim sup [A, ϕˆt,rI] = 0. t→0 r∈R k k For every ϕ BUC, • ∈ lim [A, ϕˆtI] = 0. t→0 k k We want to add another characterization of band-dominated operators. Although there is no known reference in the literature where these observations can be found in the present form, their proofs are based on well known ideas. We consider the continuous piecewise linear splines ϕ and ψ of norm one defined by 1 : x 1 1 : x 3 ϕ(x) = | | ≤ 3 , ψ(x) = | | ≤ 4 , 0 : x 2 0 : x 4 ( | | ≥ 3 ( | | ≥ 5 and their inflated and shifted copies ϕα,R and ψα,R, where f α,R(x) := f(x/R α). − Proposition 1.56. The mapping S : (lp, ) (lp, ), given by R L P → L P α,R α,R SR(A) := ϕˆ Aψˆ I, (1.18) ∈ αXZ is well defined and linear. The series (1.18) converges -strongly, and SR(A) is a band operator p P with SR(A) 2 A for every A (l , ) and R N. k k ≤ k k ∈ L P ∈ p ∞ Proof. For the spaces l with 1 p < and for (l )0 see [63], Proposition 2.2.2 or [44], ≤ ∞ ∞ ∞ Lemma 3.14. In order to prove the -strong convergence on l we take SR(A) ((l )0, ) P ∞ ∈∞ L P and note that the sequence (SR(A)Pn)n∈N has a -strong limit in (l , ) since l is complete w.r.t. -strong convergence, and by the argumentsP of the 9th stepL in theP proof of Theorem 1.27 on pageP 20. Since χF SR(A)χGI = 0 for arbitrary subsets F,G of Z with dist(F,G) 2R, we find that SR(A) is banded (see [44], Proposition 1.36). ≥ 36 PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

Notice that in case of a band operator A, the equality SR(A) = A holds for sufficiently large R due to the fact that ϕˆα,RAψˆα,RI =ϕ ˆα,RA for all α and large R. Let A be a band-dominated operator. Then, by definition, there is a sequence (Am) of band operators which tends to A in the norm. One might assume that such Am can be constructed by taking the restrictions of the matrix representation of A to a finite number of diagonals. In general, this is not true, as [44], Remark 1.40 shows. Nevertheless, the sequence (SR(A))R∈N provides a canonical approximation of A by band operators, in analogy to the Fejér-Cesàro means for the approximation of continuous functions by trigonometric polynomials. Indeed, for every fixed m and sufficiently large R,

S (A) A S (A Am) + S (Am) Am + Am A 3 Am A , k R − k ≤ k R − k k R − k k − k ≤ k − k where the latter becomes as small as desired if m is chosen suﬃciently large. p Conversely, if for an operator A (l ) the band operators SR(A) exist and converge to A in the norm, then A is band-dominated.∈ L Thus we have

Corollary 1.57. An operator A (lp) is band-dominated, if and only if the operators S (A) ∈ L R exist for every R N and SR(A) A 0 as R . ∈ k − k → → ∞

Concerning the stability of a finite section sequence (An) = (PnAPn +Qn) of A lp we already know that the invertibility of A is a necessary condition.19 One may say that the∈ singleA operator A, which is just the -strong limit of (An), captures one facet of the asymptotic behavior of this sequence. This seemsP to be a quite facile observation at a first glance, but nevertheless this is basically what we are going to do for the complete characterization of stability: Take snapshots of (An) by passing to certain limit operators and draw on their invertibility. Of course, this simplest snapshot A describes the behavior at the center of the truncation intervals [ n, n]. The much more interesting points where the projections Pn have their major impact are at− the boundary n, n , and it seems to be natural to observe the behavior there as well. To do {− } 1 2 this one might replace (An) by the shifted copies (An) := (VnAnV−n) and (An) := (V−nAnVn) which “fix one endpoint of the truncation intervals at the origin”. Figure 1.2 illustrates the movement of the focus. Clearly, the stability is invariant under these translations, and it is also 1 2 obvious that every subsequence of a stable sequence is stable again. So, if (An), (An), or at least certain subsequences of them converge -strongly then the invertibility of these limits is also P necessary for the stability of the sequence (An). Now, this immediately imposes the question if, with these additional snapshots, we have enough for the characterization of the stability of (An), in the sense that their simultaneous invertibility gives a sufficient condition, or if we need any more. The answer will be “Yes, under a natural condition, it suffices to observe the sequence at its center and at the boundaries of the truncation interval.” Also for the asymptotics of the pseudospectra of An we already know from the basic example A = V (see Section 1.4.4) that the operator A alone is not enough to capture the limiting set, and we may guess that the additional snapshots could bring more light into the darkness. Incited by these problems we are going to examine the notion of rich operators in the next step, and to elaborate this sequence algebra approach in Part2, of course in a more general and more abstract setting. There, we also discuss a Fredholm theory for sequences which extends the connection between stability and invertibility to certain connections between “almost stability” of a sequence and the Fredholm properties of its snapshots. Part3 will treat also the question on spectral approximation.

19Cf. Proposition 1.29 1.5. EXAMPLE: OPERATORS ON LP -SPACES 37

An VnAnV n −

Pm Pm

Figure 1.2: -strong limits of the sequence (A ) and its shifted copy (V A V− ). P n n n n

1.5.2 Rich operators We already discovered that it might be fruitful to examine the asymptotics of shifted copies p (V−gn AVgn ) of A (l , ), where gn is a sequence of integers whose absolute values tend to inﬁnity. If such a∈ sequence L P converges -strongly (let’s say, to A ) then we call A the limit P g g operator of A w.r.t. g. The set σop(A) of all limit operators of A is referred to as its operator spectrum. Further, we say that A is a rich operator if every sequence g of integers whose absolute values tend to has a subsequence h such that Ah exists. The theory of limit∞ operators has been intensively studied during the last years and has a wide range of applications. We only touch this concept here but, nevertheless, we state some of its highlights. For a nice introduction and a comprehensive discussion consult the work of Lindner (e.g. [44], [15]) or the fundamental book of Rabinovich, Roch and Silbermann [63].

Theorem 1.58. Let A lp be rich. Then A is -Fredholm if and only if all limit operators of A are invertible and their∈ A inverses are uniformlyP bounded.

This theorem has a long history and we particularly mention the pioneering paper [42] of Lange and Rabinovich. The proof of the if part is based on a construction of a -regularizer, which goes back to Simonenko [86] and can be found in [62] and [44], for example.P 20 The only if part was discussed in [62] and [63], Theorem 2.2.1 (for 1 < p < ), in [44] (all p and with an additional assumption on the existence of a predual setting in the∞ case p = ), and in [15], Theorem 6.28 (all p). We mention that the present approach provides this implication∞ even for every rich operator A (lp, ): ∈ L P

Proof. Let g be such that Ag exists and let B be a -regularizer for A. It is quite obvious from the deﬁnition that the operator spectrum of every -compactP operator K is trivial: σ (K) = 0 . P op { } Thus, besides V−gn AVgn Ag, we also have V−gn (AB I)Vgn 0 and V−gn (BA I)Vgn 0 -strongly. Then, for every→ T (lp, ) − → − → P ∈ K P T = V IV T V BV V AV T + V (I BA)V T , k k k −g(n) g(n) k ≤ k −g(n) g(n)kk −g(n) g(n) k k −g(n) − g(n) k and consequently (for a certain constant D > 0 independent of g, and n ) → ∞ T D A T for all T (lp, ). k k ≤ k g k ∈ K P p The dual estimate T D TAg for all T (l , ) follows analogously. Due to the - k k ≤ k k ∈ K P −1 P dichotomy, this implies the invertibility of Ag and Theorem 1.14 yields that Ag belongs to (lp, ). L P 20 Actually, also our mappings SR are inspired by this construction. 38 PART 1. BOUNDED LINEAR OPERATORS ON BANACH SPACES

p p For every y (l )0 there is a projection Ry (l , ) with norm not greater than 2 onto span y (by∈ Proposition 1.33), and therefore∈ the K ﬁrstP of the above estimates yields for the { } −1 p −1 −1 −1 p operator Ag Ry (l , ) that Ag Ry D AgAg Ry 2D. Hence, Ag (l )0 2D and ∈ K −1 P k k ≤ k k ≤ k | k ≤ with (1.6) we get Ag 2D with D independent of g. This yields the uniform boundedness of the inverses. k k ≤ We also note that every limit operator of a band-dominated operator is again band-dominated (cf. [44], Proposition 3.6).

Proposition 1.59. The set $(lp, ) of all rich operators is a closed subalgebra of (lp, ) and contains (lp, ) as a closedL ideal.P Every -regularizer of a rich -Fredholm operatorL isP rich. In particular,K P$(lp, ) is inverse closed in P (lp, ) and (lp).21 P L P L P L Proof. It is obvious from the deﬁnition that $(lp, ) is a subalgebra of (lp, ). Now, let (Ak) $(lp, ) be a sequence which tends uniformlyL P to an operator A (Llp, )P, and let h be a sequence⊂ L of integersP whose absolute values tend to . Choose a subsequence∈ L gP1 h such that 1 2 1 2 ∞ 3 2 ⊂ 3 Ag1 exists, a subsequence g g such that Ag2 exists, a subsequence g g for A , and so ⊂ n ⊂ on. Then deﬁne a new sequence g = (gn) by gn := gn. Evidently, g is a subsequence of h and all m m limit operators Ag , m N exist. One now easily checks that (Ag )m converges uniformly, and ∈ $ p its limit Ag is the limit operator of A w.r.t. g. Thus, (l , ) is closed. Let A $(lp, ) be -Fredholm, B be a -regularizer,L andP g such that A exists. From the ∈ L P P P g previous proof we already know that Ag is invertible, and we claim that Bg exists and equals −1 Ag . Then it is clear that B is rich. Indeed, since

V BV A−1 −g(n) g(n) − g = V BV (I V AV A−1) + V (BA I)V A−1 −g(n) g(n) − −g(n) g(n) g −g(n) − g(n) g = V BV [A V AV ]A−1 + V (BA I)V A−1 −g(n) g(n) h − −g(n) g(n) g −g(n) − g(n) g −1 p we get (V−g(n)BVg(n) Ag )T 0 for every T (l , ). Analogously, we can show that T (V k BV A−1−) 0 andk → obtain the -strong∈ K convergenceP of V BV to A−1. k −g(n) g(n) − g k → P −g(n) g(n) g Corollary 1.60. If dim X < then every band-dominated operator on lp = lp(Z,X) is rich. ∞ ∞ Proof. It suﬃces to consider operators aI of multiplication by a = (an) l (Z, (X)), since the shift operators are obviously rich, and $(lp, ) is a closed algebra. ∈ L With k := dim X and a ﬁxed basis in XLwe seeP that the matrix representations of the (uniformly bounded) operators an are k k-matrices and from the Bolzano-Weierstrass theorem we deduce the existence of a convergent× subsequence.

For general X we have Theorem 1.61. (see [63], Theorem 2.1.16) p The operator aI of multiplication by a = (an) l (Z, (X)) is rich if and only if the set of its ∈ L values an : n Z is relatively compact (with respect to the operator norm on (X)). { ∈ } L

21The notion of rich operators can be introduced in much more general settings L(X, P), where a certain family of “shift-like” operators is available (see [63], Section 1.2 for details). If X has the P-dichotomy then this proposition is valid again and generalizes the results of [63], Section 1.2.1. Part 2

Fredholm sequences and approximation numbers

We now turn our attention to algebras of operator sequences A = An which have a certain asymptotic structure in common. More precisely, for every sequence{ there} shall be a family of operators W t(A) which appear as -strong limits (of the sequence A or slightly modified copies). The goal of Sections 2.1 and 2.2 willP be a Fredholm theory for such sequences, which provides a couple of relations between the properties of A (such as stability or Fredholmness), properties of its entries An (e.g. the asymptotic behavior of the approximation numbers or indices), and the Fredholm properties of its so-called snapshots W t(A). Section 2.3 is devoted to a notion of Fredholm sequences which does not require any asymptotic structure and which can be regarded as a generalization of the first concept. Finally, we deal with sequences which are somehow in between of these two approaches, so-called rich sequences. They do not have an asymptotic structure as considered in the first section, but they have at least “sufficiently many” subsequences of that structured type. We apply the established tools to band-dominated operators in Section 2.5.

Such concepts of Fredholmness for operator sequences have their origin in the consideration of the finite section method for Toeplitz operators [83], are well sophisticated in the framework of Standard Algebras [30], that are classes of C∗-algebras, have been studied in the case of matrix sequences A with -strong limits W t(A) [75] and there are also variants for special classes of operator sequences∗ in a Banach space context, again with the classical notions of compactness and convergence [50]. The fusion with the concept of approximate projections was first done by the author in [80] for matrix and in [81] for operator sequences. Clearly, these latest releases are heavily influenced by its forerunners and include or adapt several former ideas. The reader will particularly rediscover the golden thread of [75] in the presentation of the Sections 2.1 and 2.2. The content of the Sections 2.1 to 2.3 and 2.5 is subject of the mentioned publications [80], [81], but here in a complete and stand-alone form, whereas the notion of rich sequences appears here for the first time and generalizes the notion of rich operators 1 and concrete tools for band- dominated operators, in a sense.

1Cf. Section 1.5.2. 40 PART 2. FREDHOLM SEQUENCES AND APPROXIMATION NUMBERS

2.1 Sequence algebras

Let (En) be a sequence of Banach spaces and let Ln stand for the identity operator on En, 2 respectively. We denote by the set of all bounded sequences An of bounded linear operators A (E ). One easily veriﬁesF that, provided with the operations{ } n ∈ L n α A + β B := αA + βB , A B := A B , { n} { n} { n n} { n}{ n} { n n} and the supremum norm A F := sup A < , becomes a Banach algebra with k{ n}k n k nkL(En) ∞ F identity I := Ln . The set { } := G : G 0 as n G {{ n} ∈ F k nkL(En) → → ∞} forms a closed ideal in . F Further, let T be a (possibly inﬁnite) index set and suppose that, for every t T , there is a Banach space Et with a uniform approximate identity t such that Et has the∈ t-dichotomy, t t t P t t P and a bounded sequence (Ln) of projections Ln (E , ) tending -strongly to the identity It on Et. Set ∈ L P P t t c := sup L L(Et) : n N < for every t T. {k nk ∈ } ∞ ∈ Suppose that, for every t T , there is a sequence (Et ) of algebra isomorphisms ∈ n Et : (im Lt ) (E ), n L n → L n −t t −1 such that (for brevity, we write En instead of (En) )

t t −t M := sup E , E : n N < . (I) {k nk k n k ∈ } ∞ T We denote by the collection of all sequences A = An , for which there exist operators tF { } ∈ F W t(A) (E , t) for all t T such that ∈ L P ∈ (t) −t t t t A := E (An)L W (A) -strongly. n n n → P These limits are uniquely determined and with the help of Theorem 1.19 it is easy to show that T is a closed subalgebra of which contains the identity and the ideal . Both, the mappings F t F G Et and W t : T (E , t), A W t(A) are unital homomorphisms for every t T . n F → L P 7→ ∈ Roughly speaking, the mappings Et allow us to transform a given sequence A T and to n ∈ F generate snapshots W t(A) from diﬀerent angles which outline several aspects of the asymptotic behavior of A. In what follows, we will examine the connections between the properties of A and the properties of its “snapshots at inﬁnity”.

Remark 2.1. The results and proofs of the subsequent sections remain true, if for some or all t t T the sequence (Ln) of projections converges -strongly to the identity, and if we replace ∈t-Fredholmness by Fredholmness, t-compactness∗ by compactness, and t-strong convergence byP -strong convergence. An approximateP projection t is not needed in thisP case. Thus,∗ the results for so-called Standard Algebras (seeP [30]) and for Banach algebras of matrix sequences in [75] are completely covered by the present considerations. Furthermore, the generalization of the approach of [75] to sequences of operators acting on inﬁnite dimensional Banach spaces in [50] is a special case of the theory in the present text as well.

2We continue to use (·) for sequences of elements in one common space, whereas the sequences in F which consist of elements coming from diﬀerent spaces En in general are denoted by {·}. 2.2. T -FREDHOLM SEQUENCES 41 J

. An . (E ) An+1 L n ... (E ) L n+1 t1 ... En Et1 n+1 ... t2 En ...... (t1) (t2) An An n n t1 t2

t P t P (E 1 ) W t1 (A) (E 2 ) W t2 (A) L L T Figure 2.1: Snapshots of a sequence A = An . { } ∈ F

2.2 T -Fredholm sequences J Here we introduce the notion of Fredholm sequences. A sequence will be called Fredholm, if it is “invertible modulo an ideal of compact sequences”, where in our ﬁrst approach these “compact sequences” will be generated by lifting t-compact operators. For this, it turns out to be reasonable to suppose that the perspectivesP from which a sequence can be looked at, and which are given by the homomorphisms W t, are diﬀerent from each other. Therefore, and in all what follows, we suppose that the separation condition

Kt if t = τ W τ Et (Lt Kt) = (II) { n n } 0 if t = τ ( 6 holds for all τ, t T and every Kt t := (Et, t). Metaphorically speaking, this means that the directions∈ from which one can∈ K look atK a sequenceP are separated in the sense that the t t t t t -compact operators K and their liftings En(LnK ) which arise from one point of view t T areP invisible from every other direction. { } ∈

Deﬁnition 2.2. We put 3

t := Et (Lt Kt) + G : Kt t, G 0 ( t T ), J {{ n n } { n} ∈ K k nk → } ∀ ∈ m T ti ti ti := closF T J : m N, ti T, J . J { n } ∈ ∈ { n } ∈ J (i=1 ) X One may refer to the elements of these sets as compact sequences. Actually, we will see that this is entirely justiﬁed since the algebraic structure is similar to the notions of compact or -compact operators. We start with the following observation. P

Proposition 2.3. All t, t T as well as T are closed ideals in T . J ∈ J F 3The consideration of such ideals goes back to the approach of Silbermann for the ﬁnite section method applied to Toeplitz operators in [83]. See also Theorem 2.17 and the annotations there. 42 PART 2. FREDHOLM SEQUENCES AND APPROXIMATION NUMBERS

t T Proof. For t T , K and A = An we have ∈ ∈ K { } ∈ F t t t (t) t t t t t (t) t AnE (L K) = E (A K) = E (L W (A)K) + E (L (A W (A))K), n n n n n n n n n − t t t t (t) t t t t t (t) t E (L K)An = E (L KA ) = E (L KW (A)) + E (L K(A W (A))), n n n n n n n n n n − t where W t(A) (E , t) and hence W t(A)K,KW t(A) t. Due to the Condition (I) and (t) ∈t L P t ∈ K since (An W (A)) tends -strongly to 0, the last summands tend to zero in the norm as n in both− cases. Consequently,P all t as well as T are ideals in T , where T is closed by→ its ∞ deﬁnition. J J F J k t t t k t Let ( Jn )k∈N = ( En(LnKk) + Gn )k∈N be a Cauchy sequence. From Theorem 1.19 { } t{ t t } { } ⊂ J we obtain (with C := c M BPt )

Kt Kt = W t Et (Lt (Kt Kt)) + Gk Gl k k − l k k { n n k − l } { n} − { n} k Ct ( Et (Lt Kt ) + Gk ) ( Et (Lt Kt) + Gl ) ≤ k { n n k } { n} − { n n l } { n} k = Ct J k J l , k{ n } − { n}k t t t therefore the sequence (Kk)k∈N is a Cauchy sequence in , and since is closed it possesses a limit Kt t. Analogously, the estimate K K ∈ K Gk Gl ( Et (Lt Kt ) + Gk ) ( Et (Lt Kt) + Gl ) + Et (Lt (Kt Kt )) k{ n} − { n}k ≤ k { n n k } { n} − { n n l } { n} k{ n n l − k }k J k J l + M tct Kt Kt ≤ k{ n } − { n}k k l − kk k shows that Gn converges to a certain Gn . Now it’s easy to see that the sequence Et (Lt Kt) {+ }G t is the limit of ({J k }) ∈and G the closedness of t is proved. { n n } { n} ∈ J { n } k J Definition 2.4. A sequence A T is said to be T -Fredholm, or regularizable with respect ∈ F J to T , if the coset A + T is invertible in the quotient algebra T / T . J J F J Notice that this property depends on the underlying algebra T and the ideal T . It is obvious that the set of T -Fredholm sequences is open in T , theF sum of a T -FredholmJ sequence and a sequence fromJ the ideal T is T -Fredholm andF that the productJ of two T -Fredholm sequences is T -Fredholm again.J WeJ will see that there are much deeper similaritiesJ with the usual FredholmJ property of operators. In particular, there are analogues to the kernel and cokernel dimension as well as the index. Let us start with a basic result on the regularization of a T -Fredholm sequence. J T T Proposition 2.5. Let A be -Fredholm. Then there exist finite subsets t1, ..., tm and τ , ..., τ of T and a δ >∈0 Fsuch thatJ the following holds: { } { 1 l} For each A˜ T with A A˜ < δ there are sequences B, C T and G, H as well as ∈ F k − k ∈ F ∈ G operators Kti ti and Kτi τi such that ∈ K ∈ K m t t t BA˜ = I + E i (L i K i ) + G, (2.1) { n n } i=1 X l τ τ τ AC˜ = I + E i (L i K i ) + H. (2.2) { n n } i=1 X Proof. By the definitions of T -Fredholmness and of the ideal T , there exist a sequence Aˆ T , J Jˆti ti ˆ τi τi ∈ F finite subsets t1, ..., tm and τ1, ..., τl of T and sequences J , K as well as { } { } ∈ J ∈ J 2.2. T -FREDHOLM SEQUENCES 43 J

Jˆ, Kˆ T with Jˆ , Kˆ < 1/4 such that ∈ J k k k k m l t τ AAˆ = I + Jˆ i + Jˆ and AAˆ = I + Kˆ i + Kˆ . i=1 i=1 X X For A˜ T with A˜ A < δ := 1/(4 Aˆ ) we have ∈ F k − k k k m t AÂ˜ = I + Jˆ i + Jˆ + Aˆ(A˜ A) − i=1 X where Jˆ + Aˆ(A˜ A) < 1/2. Hence, the sequence I + Jˆ + Aˆ(A˜ A) is invertible in the Banach algebrak T and− we cank define − F −1 T B := [I + Jˆ + Aˆ(A˜ A)] Aˆ and − ∈ F t −1 t t J i := [I + Jˆ + Aˆ(A˜ A)] Jˆ i i − ∈ J for i = 1, ..., m. Due to the definition of the ideals t, there are operators Kti ti and a J ∈ K sequence G such that ∈ G m m t t t t BA˜ = I + J i = I + E i (L i K i ) + G. { n n } i=1 i=1 X X Analogously, one obtains sequences C T and H as well as operators Kτi τi such that Eq. (2.2) holds. ∈ F ∈ G ∈ K

Applying the homomorphisms W t, t T to the Equations (2.1) and (2.2), and utilizing the separation condition, we see that the following∈ theorem is in force.

Theorem 2.6. If a sequence A T is T -Fredholm, then all corresponding operators W t(A) are t-Fredholm on Et and the number∈ F ofJ the non-invertible operators among them is finite. P 2.2.1 Systems of operators For fixed t T let P t t be a t-compact operator. We set Pˆt := It P t and obtain ∈ ∈ K P − It (It Lt )P t = Lt P t + Pˆt. − − n n t t t t t t t t Since P , there is an nt N, such that (I Ln)P , P (I Ln) < 1 for all n nt. Thus, for∈n K n , we obtain the∈ invertibility ofkIt −(It Ltk)Pk t and,− moreover,k the inverses≥ are ≥ t − − n ∞ (It (It Lt )P t)−1 = ((It Lt )P t)k (Et, t). − − n − n ∈ L P kX=0 This justifies the following definition for n n : ≥ t ∞ ∞ It = Lt P t ((It Lt )P t)k + Pˆt ((It Lt )P t)k . n − n − n kX=0 kX=0 =:P t ˆt n =:Pn | {z } | {z } 44 PART 2. FREDHOLM SEQUENCES AND APPROXIMATION NUMBERS

For the operators P t we have im P t im Lt and P t P t tends to 0 as n since n n ⊂ n k − nk → ∞ P t P t = P t It + It P t = Pˆt Pˆt k − nk k − − nk k n − k ∞ t t t ˆt t t t k ˆt (I Ln)P = P ((I Ln)P ) P k −t t kt 0. − ≤ k k 1 (I Ln)P → k=1 − k − k X

Applying the homomorphisms Et , we can lift them to a sequence Rt T by the rule n { n} ∈ F t t t En(Pn): n nt Rn := ≥ . (0 : n < nt

Indeed, it is easy to check that Rt is bounded, and it belongs to t because of { n} J Rt Et (Lt P t) = Et (P t Lt P t) M t( P t P t + (It Lt )P t ) 0. k n − n n k k n n − n k ≤ k n − k k − n k → t t t One can further show that, for n large, dim im Rn = dim im Pn dim im P and, for every k < dim im P t, there is an N t with dim im Rt > k whenever n N t≤. k n ≥ k t t t ˆt t ˆt t Deﬁnition 2.7. For t T let P . The system (P , P ,Pn, Pn,Rn) of operators (with n n ), which we can construct∈ as above,∈ K is called a P t-corresponding system of operators. ≥ t The following proposition shows that for sequences A T , applying such P t-corresponding ∈ F systems, certain properties of boundedness of the operators W t(A) can be devolved on the sequence A. T t t t t Proposition 2.8. Let A = An , P be a -compact operator with the P - t ˆt { t ˆ}t ∈t F ∈ K P corresponding system (P , P ,Pn, Pn,Rn). Then

t t t t t t t t lim sup AnRn M W (A)P , and lim sup RnAn M P W (A) . n→∞ k k ≤ k k n→∞ k k ≤ k k Proof. From the above considerations we get

A Rt = A Et (P t ) = Et (A(t)P t ) M t A(t)P t k n nk k n n n k k n n n k ≤ k n nk t (t) t t t t t t t t M ( A P + M c A P P ) M W (A)P , ≤ k n k k kk n − k → k k (t) t t t t since An W (A) -strongly and P is -compact. The dual assertion can be checked analogously.→ P P

Moreover, if P t is a projection, then the relations

∞ ∞ PˆtPˆt = PˆtPˆt ((It Lt )P t)k = Pˆt ((It Lt )P t)k = Pˆt , n − n − n n k=0 k=0 ∞X X Pˆt Pˆt = Pˆt ((It Lt )P t)kPˆt = Pˆt ItPˆt + 0 = Pˆt, and n − n kX=0 ˆt ˆt ˆt ˆt ˆt ˆt ˆt ˆt ˆt ˆt ˆt PnPn = Pn(P Pn) = (PnP )Pn = P Pn = Pn

t t t imply that also the operators Pn and Rn (En) are projections and one can easily check t t ∈ K t ∈ L that dim im Rn = dim im Pn = dim im P for suﬃciently large n. 2.2. T -FREDHOLM SEQUENCES 45 J

t 2.2.2 The approximation numbers of An and the snapshots W An { } T Here is a further connection between the operators An of a sequence An and its snapshots W t A . { } ∈ F { n} T ti Theorem 2.9. Let A = An , m N and t1, . . . , tm T be such that all W (A) are ti r { } ∈ F ∈ m ∈ti l -Fredholm. Then sk(An) 0 for all k i=1 dim ker W (A), and sk(An) 0 for all P m → ≤ → k dim coker W ti (A) as n . ≤ i=1 → ∞ P If, for one t T , the snapshot W t(A) is properly t-deﬁcient from the right (from the left) then r P ∈ l P s (An) 0 (or s (An) 0, respectively) for every k N as n . k → k → ∈ → ∞ Proof. Proposition 1.32 reveals that, if all of these W ti (A) are normally solvable then, for each ti ti i = 1, ..., m and every respective non-negative integer ki dim ker W (A), we can ﬁx a - i ≤ ti P compact projection P onto a ki-dimensional subspace of ker W (A) and choose the respective system (P t, Pˆt,P t , Pˆt ,Rt ) of projections. Moreover, for each i and every n max n we n n n ≥ j tj choose a normed basis xn ki of im Ri , such that for arbitrary scalars αn the following holds: { i,l}l=1 n i,j

ki αn αn xn for all p = 1, ..., k . (2.3) | i,p| ≤ i,j i,j i j=1 X

It is a simple consequence of Auerbach’s Lemma (see Proposition 1.4) that such a basis always

j j j tj exists. Let i = j. Due to the separation condition (II), since Pn P 0 and P , we have 6 k − k → ∈ K

Ri Rj = Eti (P i )Etj (P j) M tj E−tj (Eti (P i ))P j 0. (2.4) k n nk k n n n n kL(En) ≤ k n n n nk → We now prove that there is a number N N, such that, for all j = 1, ..., m, k = 1, ..., kj, n N, all scalars α , and all y Y := m ker∈ Ri ≥ i,l ∈ n i=1 n m ki T n ti i αj,k γ αi,lxi,l + y , where γ = 2 max M P . (2.5) | | ≤ i=1,...,m k k i=1 l=1 X X In other words, all of these xn are linearly independent in a “uniform” manner. i,j By (2.4), for every (0, 1), there exists a number N N such that ∈ ∈ m i j i ti i ti i kj R R and R M P (1 + )M P (2.6) k n nk ≤ k nk ≤ k nk ≤ k k 6 j=1X,j=i for all i = 1, ..., m and n N. Now let αi,l be arbitrary but ﬁxed scalars and choose i0, l0 such that α = max α ≥. Thanks to (2.3) and (2.6) we ﬁnd (since xn = Ri xn ) | i0,l0 | i,l | i,l| i,j n i,j

m ki ki0 m ki i0 n i0 n i0 n Rn αi,lxi,l + y αi0,lRn xi ,l αi,lRn xi,l k k ≥ 0 − i=1 l=1 l=1 i=1,i=6 i0 l=1 X X X X X

ki0 m ki

α xn α Ri0 Ri xn ≥ i0,l i0,l − | i,l|k n n i,lk l=1 i=1,i=6 i0 l=1 X X X m

α α k Ri0 Ri ≥ | i0,l0 | − | i0,l0 | ik n nk 6 i=1X,i=i0 (1 ) α . ≥ − | i0,l0 | 46 PART 2. FREDHOLM SEQUENCES AND APPROXIMATION NUMBERS

Thus, for all n N, for all j = 1, ..., m, and all k = 1, ..., k , ≥ j m k Ri0 i α α k n k α xn + y | j,k| ≤ | i0,l0 | ≤ 1 i,l i,l i=1 − X Xl=1 k 1 + m i ti i n max M P αi,lxi,l + y . ≤ 1 i=1,...,m k k · − i=1 l=1 X X

Choosing = 1/3 gives assertion (2.5). Obviously, En decomposes into the direct sum

E = span xn ... span xn Y n { 1,1} ⊕ ⊕ { m,km } ⊕ n for n N, and we can introduce functionals f n E∗ by the rule ≥ i,j ∈ n

m kk n n n n fi,j αk,lxk,l + y := αi,j 1 i m, 1 j ki. ! ≤ ≤ ≤ ≤ kX=1 Xl=1 n n Then we always have fi,j γ and fi,j(Yn) = 0 . Now, we denote by Rn (En) the linear operators k k ≤ { } ∈ L m ki n n Rnx := fi,j(x)xi,j. i=1 j=1 X X m The operators Rn are projections of rank dim im Rn = k := i=1 ki and they are uniformly bounded with respect to n. Moreover, for every x En we have ∈ P

m ki m ki A R x = A f n (x)xn f n (x) A Ri xn k n n k n i,j i,j ≤ | i,j |k n n i,jk i=1 j=1 i=1 j=1 X X X X

m ki m γ x A Ri x n = γ x k A Ri . ≤ k kk n nkk i,jk k k ik n nk i=1 j=1 i=1 X X X Since A Ri 0 for each i (see Proposition 2.8) it follows that k n nk → sr (A ) = inf A + F : F (E ), dim ker F k k n {k n k ∈ L n ≥ } i An An(Ln Rn) = AnRn γk max AnRn 0. ≤ k − − k k k ≤ i k k →

Due to Proposition 1.32 we can also choose ti -compact projections P˜ti with P˜ti W ti (A) = 0 P ˜ ti n ˜n ˜i ˜ and proceed in the same way to ﬁnd the analogues ki dim coker W (A), x˜i,j, fi,j, Rn and Yn, ˜ ≤ ˜ ˜ m ˜ and to construct a bounded sequence (Rn) of projections Rn, being of rank k = i=1 dim ki. Then P

m k˜i m k˜i R˜ A x = fñ (A x)˜xn fñ (R˜i A x) + fñ ((I R˜i )A x) xñ k n n k i,j n i,j ≤ | i,j n n i,j − n n |k i,jk i=1 j=1 i=1 j=1 X X X X ˜ m ki fñ R˜i A + fñ (I R˜i ) A x , ≤ k i,jkk n nk k i,j − n kk n|| k k i=1 j=1 X X 2.2. T -FREDHOLM SEQUENCES 47 J where fñ (I R˜i ) 0 as n due to the following observations: for each y Y˜ we have k i,j − n k → → ∞ ∈ n fñ ((I R˜i )y) = fñ (y) = 0, that is fñ (I R˜i )(I R˜ ) = 0, and further i,j − n i,j i,j − n − n

m k˜s fñ (I R˜i )R˜ fñ fñ (I R˜i )˜xn 0 k i,j − n nk ≤ | s,l|k i,j − n s,lk → s=1 X Xl=1 ˜i n ñ ˜i n ñ ˜i n ñ ˜i ˜s n since (I Rn)˜xi,l = 0 and fi,j(I Rn)˜xs,l = fi,jRnx˜s,l fi,j RnRn x˜s,l 0 for s = i as n −. Thus, we againk obtain− k k k ≤ k kk kk k → 6 → ∞ sl (A ) R˜ A 0 as n . k n ≤ k n nk → → ∞ Now, suppose that one operator W t(A) is properly t-deficient from the right (or left). Then P for each k N and each > 0 there is a projection Q t, rank Q = k such that W t(A)Q < ∈ ∈ K k k (or QW t(A) < , respectively). Choosing a system of projections w.r.t. Q, we deduce again k k r l from Proposition 2.8 that lim supn sk(An) = 0 (or lim supn sk(An) = 0) since can be chosen arbitrarily small. Finally notice that W ti (A) being ti -Fredholm but not normally solvable implies ti -deficiency from both sides by Proposition 1.32P . P

2.2.3 Regular T -Fredholm sequences J Definition 2.10. We introduce the nullity α(A) and deficiency β(A) of a sequence A T by ∈ F t t α(A) := dim ker W (A) and β(A) := dim coker W (A). ∈ ∈ Xt T Xt T A T -Fredholm sequence A T is said to be regular, if all operators W t(A) are Fredholm (in J ∈ F the usual sense). Of course, due to Theorem 2.6, A is regular if and only if α(A) and β(A) are finite, hence we are in a position to introduce the index of a regular sequence A by 4

ind A := α(A) β(A). − Applying Proposition 2.5 and the well known properties of Fredholm operators as stated in Theorem 1.2 it is not hard to prove the following result.

Proposition 2.11. Let A T be a regular T -Fredholm sequence and B T . ∈ F J ∈ F If B is suﬃciently small then α(A + B) α(A), β(A + B) β(A) and • k k ≤ ≤ ind(A + B) = ind(A).

If B T has only compact snapshots then A + B is regular and ind(A + B) = ind A. • ∈ J If B then α(A + B) = α(A) and β(A + B) = β(A). • ∈ G If B T is also a regular T -Fredholm sequence then ind(AB) = ind A + ind B. • ∈ F J Observe that, if all approximate identities t consist of ﬁnite rank operators only, then (Et, t) is a subset of (Et), and hence every T -FredholmP sequence is regular. K P K J 4This already appeared in [30] and [50]. 48 PART 2. FREDHOLM SEQUENCES AND APPROXIMATION NUMBERS

2.2.4 The splitting property and the index formula

T T Proposition 2.12. Let A = An be a -Fredholm sequence and suppose that all of { } ∈ F J its snapshots W t(A) are properly t-Fredholm operators, respectively. If α(A) is finite then lim inf sr (A ) > 0, and if β( P) is finite then lim inf sl (A ) > 0. n α(A)+1 n A n β(A)+1 n Proof. Assume that α(A) is finite. Due to Equation (2.1) m t t t BA = I + E i (L i K i ) + G { n n } i=1 X t from Proposition 2.5, all operators W (A) with t T t1, ..., tm have kernels of dimension zero. ∈ \{ } t Moreover, for every i = 1, ..., m, Corollary 1.31 provides an operator Bi (E i , ti ) such that ∈ L P P i := Iti BiW ti (A) ti is a projection onto the kernel of W ti (A). Define Pî := Iti P i and − ∈ K − furthermore, for every n N, (with Bn = B) ∈ { } m D := B Eti (Lti Kti Bi). n n − n n i=1 X T t (ti) It is obvious that D and due to the i -strong convergence of An we see that { n} ∈ F P ti ti ti i ti ti ti î ti ti ti i (ti) ti E (L K B )An E (L K P ) = E (L K B (A W (A))) 0 k n n − n n k k n n n − k → m m for all i, i.e. Eti (Lti Kti Bi)A Eti (Lti Kti Pî) . Thus { i=1 n n n − i=1 n n } ∈ G P m P D A = B A Eti (Lti Kti Bi)A n n n n − n n n i=1 m X m = L + Eti (Lti Kti ) Eti (Lti Kti Pî) + H n n n − n n n i=1 i=1 Xm X ti ti ti i = Ln + En (Ln K P ) + Hn i=1 X i ti with Hn . Since dim im P = dim ker W (A) for all i, we find { } ∈ G m t t t i dim im E i (L i K i P ) α(A). n n ≤ i=1 ! X For sufficiently large n we have H < 1/2 and, applying Corollary 1.37, it follows k nk 1 ( (L + H )−1 )−1 = sr(L + H ) 2 ≤ k n n k 1 n n = inf L + H F : dim ker F 1 {k n n − k ≥ } m ti ti ti i inf Ln + Hn F + En (Ln K P ) : dim ker F α(A) + 1 ≤ ( − ≥ ) i=1 X = inf DnAn F : dim ker F α(A) + 1 {k − k ≥ } inf DnAn DnF : dim ker F α(A) + 1 ≤ {k − k ≥ } r Dn inf An F : dim ker F α(A) + 1 Dn s (An). ≤ k k {k − k ≥ } ≤ k{ }k α(A)+1 If we start with Equation (2.2) we can proceed analogously to obtain sl (A ) const > 0 β(A)+1 n for all sufficiently large n. ≥ 2.2. T -FREDHOLM SEQUENCES 49 J

We have that all operators W t(A) of a regular T -Fredholm sequence A are Fredholm and t- Fredholm, hence properly t-Fredholm by PropositionJ 1.26. In view of Theorem 2.9 this yieldsP the following result. P

T T Theorem 2.13. Let A = An be a regular -Fredholm sequence. Then the approxima- { } ∈ F J tion numbers from the right have the α(A)-splitting property, that is

r r lim s (An) = 0 and lim inf s (An) > 0. n→∞ α(A) n→∞ α(A)+1

Analogously, the approximation numbers from the left have the β(A)-splitting property.

T T Theorem 2.14. Let A = An be a regular -Fredholm sequence. Then, for suﬃciently { } ∈ F J large n, the operators An are Fredholm and their indices coincide with ind A, in other words

t lim ind An = ind W (A). n→∞ ∈ Xt T Proof. Theorem 2.13 shows that sr (A ) > 0 and sl (A ) > 0 for suﬃciently large n, α(A)+1 n β(A)+1 n hence An is Fredholm by Corollary 1.38. Moreover, for large n there is always an operator Fn l with dim coker Fn β(A) such that An Fn s (An) + 1/n. We can choose a projection ≥ k − k ≤ β(A) Sn of rank β(A) with norm less than β(A) + 2 such that SnFn < 1/n (see Proposition 1.6) and we deduce from Theorem 2.13 that k k

S A S A S (A F ) + S F 2 n n < n n n n n n n sl (A ) + 0. k k k k k − k k k β(A) n n→∞ β(A) + 2 Sn ≤ Sn ≤ n → k k k k

We set Sn := 0 for the remaining smaller n and conclude that SnAn , that is the sequence ˜ T T ˜ { } ∈ G ˜ An := (Ln Sn)An is -Fredholm with α( An ) = α(A), β( An ) = β(A) and { }˜ { − } ∈ F J { } { } ind An = ind A (cf. Proposition 2.11). Analogously, we choose a sequence Rn of { } ˜ { } ∈ F projections Rn of rank α(A) and Rn α(A) + 2 for large n, such that AnRn . k k ≤ { } ∈ G We now consider the sequence C = Cn := (Ln Sn)An(Ln Rn) and we ﬁnd that it is { } { − − } a regular T -Fredholm sequence with α(C) = α(A), β(C) = β(A) and ind C = ind A. More J precisely, there is an N N and a constant C > 0 such that for all n N ∈ ≥ sr (C ) = sl (C ) = 0 and sr (C ), sl (C ) > C, α(A) n β(A) n α(A)+1 n β(A)+1 n which proves that the Cn are Fredholm of the index α(A) β(A) = ind A. Since Ln Rn and − − Ln Sn are Fredholm of index zero, we ﬁnd that An is Fredholm of index ind A. −

Remark 2.15. Note that there is a bounded sequence D such that for sufficiently large n { n} the operator Dn is a generalized inverse for the Cn from the previous proof. Moreover, A C − belongs to , hence we find that A + is generalized invertible in / , whenever A is a regular T -FredholmG sequence. G F G J Furthermore, this shows that the nullity, deficiency and the index of a structured operator sequence A being T -Fredholm as introduced in Definition 2.10 are universal characteristics J of this sequence in the following sense: If these numbers exist for the T -Fredholm sequence A ˜ J ˜ in one setting T then they are the same in every setting T in which A is T -Fredholm. In Section 2.3 weF will recover this observation from a more generalF point of view.J 50 PART 2. FREDHOLM SEQUENCES AND APPROXIMATION NUMBERS

2.2.5 Stability of sequences A T ∈ F Deﬁnition 2.16. A sequence A = An is called stable, if there is an index n0 such that all operators A , n n , are invertible{ and} ∈ Fsup A−1 : n n < . n ≥ 0 {k n k ≥ 0} ∞ It is a well known result of Kozak [39] that a sequence A is stable if and only if the coset A+ is invertible in / . Utilizing the higher structure of the∈ F given setting, namely the existence ofG F G t-strong limits W t(A), we can prove a stronger result. This observation has a long history and isP known as the Lifting Theorem in the literature, see for example the books [12], Section 7.8ﬀ, [29], Section 1.6 or [30], Section 5.3. It has its origin in [83].

Theorem 2.17. A sequence A T is stable if and only if A is T -Fredholm and all W t(A), t T , are invertible. In particular,∈ F T / is inverse closed in / J. ∈ F G F G T t Proof. Let A = An be -Fredholm and all W (A) be invertible. Then A is regular and { } J r l Theorem 2.13 yields an N N and a constant C > 0 such that s (An) C and s (An) C for ∈ 1 ≥ 1 ≥ all n > N. With Corollary 1.37 we deduce that A is stable. T −1 Conversely, let A = An be stable and deﬁne a sequence B = Bn by Bn := A if { } ∈ F { } ∈ F n An is invertible and Bn := Ln otherwise. Then BA I, AB I . Notice that for every t T the sequence − − ∈ G ∈ (At ) := (E−t(A )Lt + (It Lt )) (Et, t) n n n n − n ∈ F P t t −t t t t is also stable, tends to W (A) and the operators Bn := En (Bn)Ln +(I Ln) are the inverses of t t − t An for suﬃciently large n. Proposition 1.29 shows that W (A) is invertible and (Bn) converges t-strongly to (W t(A))−1. Thus, B T . P ∈ F 2.3 A general Fredholm property

The notion of T -Fredholmness for structured operator sequences A T as it was introduced in the precedingJ sections depends on the underlying setting determined∈ F by T and T . On the one hand, it is convenient to work in this context because there we can describeF theJ Fredholm properties of a sequence A quite well in terms of Fredholm properties of its snapshots W t(A). On the other hand, the main disadvantage lies in the fact that a sequence which is Fredholm in one setting does not need to be Fredholm in another setting.5 Thus, in what follows, we introduce a “universal” Fredholm property for the larger framework of all bounded operator sequences in . This approach has been extensively studied in the case of C∗-algebras of operator sequences onF ﬁnite dimensional spaces in [30], Chapter 6.3. and was derived by Roch [70]. In particular, we will recover the mentioned universal characteristics α(A), β(A) and ind(A) in this larger framework again. To avoid degenerated cases we assume that lim sup dim E = . n n ∞ Deﬁnition 2.18. A sequence K is said to be of almost uniformly bounded rank if { n} ∈ F

lim sup rank Kn < . n→∞ ∞ Let denote the closure of the set containing all sequences of almost uniformly bounded rank. TheI elements of are referred to as compact sequences. I One easily checks that forms a proper closed ideal in which contains . I F G 5 t t t t t t T Clearly, A = I + {En(LnK )} with K ∈ K(E , P ) is J -Fredholm, but if we drop t from the index set T T˜ and consider A as a sequence in F with T˜ = T \{t} then we loose this property. 2.3. A GENERAL FREDHOLM PROPERTY 51

Deﬁnition 2.19. Now we are in a position to introduce a class of Fredholm sequences in by F calling A = An Fredholm if A + is invertible in / . { } ∈ F I F I Evidently, we have the typical statements as known for the classical Fredholmness

Stable sequences are Fredholm and never compact. • Products of Fredholm sequences are Fredholm. • The sum of a Fredholm sequence and a compact sequence is Fredholm. • The set of all Fredholm sequences is open in . • F If A is Fredholm, then A∗ is of Fredholm type. • { n} { n} For an equivalent characterization of Fredholm sequences we need the following deﬁnition.

Deﬁnition 2.20. Let A = An . If there is a ﬁnite number α Z+ with { } ∈ F ∈ r r lim inf sα(An) = 0 and lim inf sα+1(An) > 0, n→∞ n→∞ then this number is called the α-number of A and it is denoted by α(A). Analogously, we introduce β(A), the β-number of A, as the β Z+ with ∈ l l lim inf sβ(An) = 0 and lim inf sβ+1(An) > 0. n→∞ n→∞

Besides the well known result of Kozak [39], by applying Corollary 1.37, we immediately get the following characterization of stability in the large algebra . F Theorem 2.21. For a sequence A the following are equivalent. ∈ F A is stable. • A + is invertible in / . • G F G α(A) = β(A) = 0. • Theorem 2.22. For a sequence A the following are equivalent. ∈ F A is Fredholm. • There are sequences B1, B2 such that B1A I and AB2 I are of almost uniformly • bounded rank. ∈ F − −

A has an α-number and a β-number. • Proof. Let A = An be Fredholm. Then there are sequences D, Hi, Gi (i = 1, 2) such that { } ∈ F DA = I + H1 + G1 and AD = I + H2 + G2, where Hi are of almost uniformly bounded rank and −1 −1 Gi < 1/2. Since I + Gi are invertible, we can deﬁne B1 := (I + G1) D, K1 := (I + G1) H1 k k −1 −1 as well as B2 := D(I + G2) , K2 := H2(I + G2) . This implies the second assertion. Now let and with . For each Kn = Bn An I n0 N k := supn≥n0 rank Kn < n n0 we can introduce{ } { a projection}{ } −R with kernel∈ of dimension k and norm R∞ k + 1 such≥ that n k nk ≤ RnKn = 0, due to Proposition 1.5. Then RnBnAn = Rn. Moreover, for each n n0 we observe that sr (R ) 1, because otherwise there would exist an operator F with dim≥ ker F k + 1 k+1 n ≥ ≥ 52 PART 2. FREDHOLM SEQUENCES AND APPROXIMATION NUMBERS

such that Rn F < 1. Hence Ln Rn + F would be invertible, but since rank(Ln Rn) = k this yieldsk a contradiction.− k Thus − −

1 sr (R ) = inf R F : dim ker F k + 1 ≤ k+1 n {k n − k ≥ } = inf R B A F : dim ker F k + 1 {k n n n − k ≥ } inf R B A R B F : dim ker F k + 1 ≤ {k n n n − n n k ≥ } R B sr (A ), ≤ k nkk nk k+1 n hence α(A) k + 1 exists. Analogously we ﬁnd a β-number for A, that is, the third assertion holds. ≤ Finally, let A have an α-number and a β-number and let N N be such that ∈ inf sr (A ), sl (A ): n N > 0. { α(A)+1 n β(A)+1 n ≥ }

Then An, n N, are Fredholm operators by Corollary 1.38. In view of Equation (1.11) there are a constant≥ C > 0 and a sequence R of projections R with kernels of dimension { n} ∈ F n α(A) such that inf Anx : x im Rn, x = 1 C for all n N. We consider the restrictions A of A to{kim R kwhich∈ are injective.k k } The ≥ spaces im(≥A ) are of the codimension n|im Rn n n n|im Rn not greater than α(A) + β(A), hence they are closed and we can choose projections Sn onto im(An im Rn ), which are uniformly bounded with respect to n N. Therefore, the operators | (−1)≥ An im Rn : im Rn im Sn are invertible and their inverses An are (uniformly) bounded by C. | → (−1) For the operators Bn := RnAn Sn we conclude that

(−1) (−1) AnBn = AnRnAn Sn = AnAn Sn = Sn, B A = B A R + B A (L R ) = R A(−1)S A R + B A (L R ) (2.7) n n n n n n n n − n n n n n n n n n − n = R + B A (L R ). n n n n − n Since the latter term is of uniformly bounded rank, this proves the Fredholmness of A.

t t t Corollary 2.23. Let A = An be a Fredholm sequence and let T , as well as E , , (Ln) t { } ∈ F P and (En) be given as in Section 2.1 such that the Conditions (I), (II) are in force. Suppose that, t t t t t t for one t T , a subsequence of (En(An)Ln) converges -strongly to an operator A (E , ). Then At ∈is Fredholm. If for all t in a certain subsetPT˜ T and with respect to one∈ L commonP ⊂ subsequence of A such operators At exist then

t t dim ker A α(A), dim coker A β(A). ≤ ≤ Xt∈T˜ Xt∈T˜ Proof. The assertion immediately results from Theorem 2.9.

For a setting T and for sequences A T we get from Theorem 2.13 F ∈ F Corollary 2.24. If A T is a regular T -Fredholm sequence then A is a Fredholm sequence ∈ F J in , that is A is invertible modulo . Moreover, the numbers α(A) and β(A) in the Deﬁni- tionsF 2.10 and 2.20 are consistent. I 2.4. RICH SEQUENCES 53

2.4 Rich sequences

Recall our motivating questions on band-dominated operators from Section 1.5 and let us consider two examples on lp(Z, C).

Firstly, set A := I + K with a -compact operator K. Then (An) converges to A whereas • 1 P 2 the two shifted copies (An) and (An) converge -strongly to the identity. This can be 1 P easily seen, e.g. for (An) one has 1 An = Vn(Pn(I + K)Pn + Qn)V−n = Vn(I + PnKPn)V−n = I + VnPnKPnV−n,

and PmVnPnK , KPnV−nPm 0 as n . We already promised (without having provedk it yet!) thatk k these snapshotsk → are also→ sufficient ∞ to characterize the stability, that is we even claim that (An) is stable if and only if A is invertible. i As a second example let J be given by the rule (xi) (( 1) xi), that is its matrix • representation equals diag(..., 1, 1, 1, 1, 1, 1,...), and7→ set A−:= J+K with a -compact − − − 1 2 P K. The central snapshot is again A, but the -strong limits of (An) and (An) do not exist anymore. To achieve convergence, we have toP pass to subsequences, and this leads to two 1 snapshots for (An), namely PJP +Q and PJP +Q, where P and Q are the complementary projections and , respectively.− Similarly, 2 provides two further snapshots. χZ+ I χZ\Z+ I (An) We again announce that these five operators tell us everything about the stability. The pending proof will be given in Section 2.5.1, but for the moment we learned something very important: Instead of considering the whole sequences A in one might pass to subsequences to F attain a plainer structure. More precisely, one could fix a strictly increasing sequence g = (gn) of positive integers and pass to the algebra g of all subsequences Ag = Agn . Obviously, the theory for elements in the respective algebrasF of subsequences , T , and{ }T is analogous to Gg Fg Jg what we have already done for the basic case gn = n. We call a sequence of operators T -structured, if for every t T the respective snapshot W t( ) exists. Our second example teaches that the finite section sequences∈ of band-dominated operators· are not T -structured in general, but one could pass to subsequences which are T -structured, and hence satisfy the requirements for our Fredholm theory. Definition 2.25. We say that a sequence A is uniformly rich with respect to T (or T -rich) if each strictly increasing sequence g of positive∈ F integers has a subsequence h g such that T ⊂ Ah . ∈ Fh To repeat this definition in other words, one may say that A is T -rich if and only if every ∈ F subsequence of A has a T -structured subsequence. Of course, one cannot expect the same behavior (like the existence of splitting numbers) under these weaker assumptions, but the following theorem shows that the stability, the general Fred- holm property of Section 2.3 as well as the α- and β-numbers can still be well described in terms of snapshots. Theorem 2.26. . The set T of all T -rich sequences is a Banach algebra T T . Moreover, the • algebras RT and T / are inverse closed in and / ,F respectively.⊂ R ⊂ F R R G F F G If every T -structured subsequence of A T has a regularly T -Fredholm subsequence then • ∈ R Jh A is Fredholm and

t t α(A) = max dim ker W (Ag) , β(A) = max dim coker W (Ag) , (2.8) " ∈ # " ∈ # Xt T Xt T 54 PART 2. FREDHOLM SEQUENCES AND APPROXIMATION NUMBERS

T where the maximum is taken over all (T -structured) subsequences Ag g of A. Moreover, T ∈ F in this case, for every (T -structured) subsequence Ag of A the numbers α(Ag) and ∈ Fg β(Ag) are splitting numbers and t lim ind Agn = ind W (Ag). n→∞ ∈ Xt T For A T the following are equivalent: • ∈ R 1. A is stable. 2. Every subsequence of A is stable. 3. Every subsequence of A has a stable subsequence. 4. Every T -structured subsequence of A is stable. 5. Every T -structured subsequence of A has a stable subsequence. T 6. Every T -structured subsequence of A is h -Fredholm and all snapshots are invertible. J T 7. Every T -structured subsequence of A has a h -Fredholm subsequence and all snapshots are invertible. J Proof. The proof of T being a closed algebra T T is straightforward. R F ⊂ R ⊂ F For the second assertion assume that A = An does not have an α-number. Then for each { } r T n N there is a number jn such that jn > jn−1 and sn(Ajn ) < 1/n. Choose a h -Fredholm ∈ T r J subsequence Ah of Aj and find a splitting number for (s (Ahn )) due to Theorem 2.13, a ∈ Fh k contradiction. Thus, A has an α-number and, analogously, a β-number, and Theorem 2.22 yields the Fredholm property of A. From Theorem 2.9 or Corollary 2.23 we deduce the relations “ ” in (2.8). Furthermore, there is a sequence j tending to infinity such that lim sl (A ) =≥ 0. n α(A) jn T T Choose a subsequence Ag of Aj and pass to the -Fredholm subsequence Ah. Notice that ∈ Fg Jh Ag and Ah have the same snapshots. Then Theorem 2.13 applies to Ah and yields the equality for α(A) in (2.8). Analogously we proceed for β(A). Now, let Ag be a T -structured subsequence of the Fredholm sequence A. Then α(Ag) < . ∞ Assume that α(Ag) is not a splitting number. Then there is a subsequence Ah of Ag such that r T limn s (Ahn ) exists and is strictly positive, hence α(Ah) < α(Ag). Pass to a -Fredholm α(Ag ) Jj subsequence Aj of Ah which certainly has the splitting property with α(Aj) being the sum of the kernel dimensions of the snapshots, a contradiction. For the β(Ag) and the index formula we proceed in an analogous manner. We now turn our attention to the third assertion. If a sequence is stable then every subsequence is so, and it is obvious that 1. 2. 3. 5. and 1. 2. 4. 5.. Furthermore, 4. 6. and ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇔ 5. 7. hold (see Theorem 2.17). Statement 7. implies that A is Fredholm with α(A) = β(A) = 0 ⇔ by the second assertion. Thus, Theorem 2.21 yields the stability of A, i.e. 7. 1. ⇒ Finally, let B be a -regularizer for A and g be a sequence tending to infinity. We pass to a G T T subsequence h g with Ah h . Then Ah + h is invertible in h / h by Theorem 2.17 and ⊂ ∈ F G T F G the (unique) inverse is given by Bh + h, hence Bh . Since g was chosen arbitrarily, this G ∈ Fh shows that B T and finishes the proof of the first assertion. ∈ R Remark 2.27. We note that a T -structured subsequence Ag of a Fredholm sequence A is not necessarily T -Fredholm. To see this simply take a setting where T is not a singleton, fix Jg t T and a t-compact finite rank projection K, set T˜ := T t and consider the sequence ∈ P t t t \{ } T˜ A = An := I En(LnKLn) . Clearly, A is Fredholm and belongs to , but there is no { } − { T˜ } F subsequence Ag being -Fredholm. Moreover, the sequence B = Bn , given by Bn = An if n Jg { } is even and Bn = Ln otherwise, is Fredholm and T˜-structured, but has no splitting numbers. 2.5. EXAMPLE: FINITE SECTIONS OF BAND-DOMINATED OPERATORS 55

2.5 Example: ﬁnite sections of band-dominated operators

Let us take another look at the band-dominated operators in lp . We now give a precise deﬁnition of the sequence-algebraic framework for the ﬁnite section sequencesA of band-dominated operators.

2.5.1 Standard ﬁnite sections

Let stand for the set of all strictly increasing sequences of positive integers and, analogously, H+ − for the set of all strictly decreasing sequences of negative integers. For a given sequence H 0 ±1 h = (hn) + we set Lhn := Phn , Ehn := im Lhn , T := 1, 0, +1 , I := I, I := χZ I and ∈ H {− } ∓ E0 p 0 := l (Z,X) Lhn := Lhn E0 : (im L0 ) (E ),B B hn L hn → L hn 7→ E±1 := im I±1 L±1 := V L V hn ∓hn hn ±hn ±1 ±1 E : (im L ) (Eh ),B V±h BV∓h hn L hn → L n 7→ n n

t t t t for every n. By := (Ln) uniform approximate identities on E are given such that E have the t-dichotomy,P and the sequences (Lt ) converge t-strongly to the identities It on Et. As hn P T P usual, we let h denote the Banach algebra of all bounded sequences Ahn of bounded linear F t { t } t operators Ahn (Ehn ) for which there exist operators W Ahn (E , ) for each t T such that for n∈ L { } ∈ L P ∈ → ∞ −t t t t E (Ah )L W Ah -strongly. hn n hn → { n }P Further, we introduce the closed ideals and T in T by Gh Jh Fh := G : G 0 , Gh {{ hn } k hn k → } T := span Et (Lt KLt ) , G : t T,K t, G . Jh {{ hn hn hn } { hn } ∈ ∈ K { hn } ∈ Gh} A sequence A T is said to be T -Fredholm, if A + T is invertible in T / T . { hn } ∈ Fh Jh { hn } Jh Fh Jh

The ﬁnite section algebra A p Let stand for the algebra of all bounded sequences A F l F { n} of bounded linear operators A (E ) and let A p denote the smallest closed subalgebra of n ∈ L n F l containing all sequences L AL with some rich A p . F { n n} ∈ Al For A = An Alp and a sequence h = (hn)n∈ + let Ah denote the subsequence Ahn . { } ∈ F N ∈ H { } It is obvious, that for all A = An Alp and each h + the central snapshot exists independently from the choice of {h: } ∈ F ∈ H

0 W (A) := -lim AnLn = W (Ah) = -lim Ahn Lhn . Pn→∞ Pn→∞

Moreover, for A = LnALn , A lp being rich, there is a sequence g + such that Ag exists, { } ∈ A ∈ H and further we can choose a subsequence h g which also provides A−h, besides Ah = Ag. Thus T T ⊂ T A , and since is a closed algebra of we even ﬁnd that Alp . ∈ R R F F ⊂ R The following proposition is the key for the application of the Fredholm theory, which was introduced in the preceding sections, to the algebra p . Al 56 PART 2. FREDHOLM SEQUENCES AND APPROXIMATION NUMBERS

Proposition 2.28. Let A Alp be such that W (A) is -Fredholm and let g +. Then there ∈ F T P ∈ H is a T -structured subsequence Ah of Ag being -Fredholm. Jh

Proof. We initially check that, for every j and for each pair of operators B,C p such ∈ H+ ∈ Al that B±j and C±j exist, we have

T D := Ljn BCLjn Ljn BLjn Ljn CLjn . (2.9) { } − { }{ } ∈ Jj m m m m T For this, put Qj := Ljn−m and Pj := Ij Qj and check that Pj belongs to j . Since the { } − T m J limit operators B±j,C±j exist, we have D j . For every m the sequence DPj is contained in T ∈ F j and J m DQj B sup (I Ljn )CLjn−m k k ≤ k k n∈N k − k which can be chosen arbitrarily small by Theorem 1.55. Thus, the closedness of T implies (2.9). Jj We now show that T Ah Lhn W (A)Lhn (2.10) − { } ∈ Jh (m) for a certain subsequence h g. Note that there is a sequence (of sequences) (A ) Alp (m) ⊂ (m) ⊂ F such that A A in the norm of Alp as n , and such that all A are of the form → F → ∞

km lm (m) (m) (m) A = LnA Ln ,A lp . { ij } ij ∈ A i=1 j=1 X Y

1 (1) We choose a subsequence g of g (applying (2.9)) such that all limit operators of all Aij exist (1) 2 and such that (2.10) holds for Ag1 . Repeating this argument we can obtain a subsequence g 1 (2) (2) of g such that all limits of all Aij exist and such that (2.10) holds for Ag2 as well, and so on. 1 2 m (k) T Hence, there are sequences g g g ... g ... such that Agm m and (2.10) holds ⊃ ⊃ ⊃ ⊃ ⊃ ∈ Fg (k) n for Agm for every m and k m. Now let the sequence h = (hn) be deﬁned by hn := gn. Since ≤ (m) Ah is the norm limit of the sequence (Ah )m∈N, we easily obtain (2.10) for Ah. Finally, let B be a -regularizer for A := W (A) and note that B lp by Theorem 1.55. The P ∈ A construction of h provides the limit operators A±h, and the proof of Proposition 1.59 shows that B±h exist as well, thus we can apply (2.9) to the operators A and B. This yields the T T -Fredholmness of Lhn W (A)Lhn and hence also the -Fredholmness of Ah, by (2.10). Jh { } Jh

T Deﬁnition 2.29. Let + stand for the set of all sequences h such that Ah . HA ⊂ H ∈ Fh ±1 ±1 Corollary 2.30. Let A Alp and W (A) be -Fredholm. Then W (Ag) are -Fredholm for every g . ∈ F P P ∈ HA T Proof. Choose a subsequence h g such that Ah is h -Fredholm by the previous proposition. t t ⊂ t J Its snapshots W (Ah) equal W (Ag) and are -Fredholm, respectively (see Theorem 2.6). P

The main results Now we are in a position to adapt the general results of Part2 to this speciﬁc class of approximation sequences of band-dominated operators and we are going describe the Fredholm properties of both, the T -structured subsequences of A Alp as well as the whole ∈ F sequence A. 2.5. EXAMPLE: FINITE SECTIONS OF BAND-DOMINATED OPERATORS 57

Theorem 2.31. Let A = An Alp and g . { } ∈ F ∈ HA ±1 If W (A),W (Ag) are Fredholm, then • +1 −1 lim ind Agn = ind W (A) + ind W (Ag) + ind W (Ag) n→∞

and the approximation numbers from the right/left of the operators Agn of Ag have the α-/β-splitting property with

+1 −1 α = dim ker W (A) + dim ker W (Ag) + dim ker W (Ag), +1 −1 β = dim coker W (A) + dim coker W (Ag) + dim coker W (Ag).

±1 If one of the operators W (A), W (Ag) is not Fredholm then, for each k N, we have • r l ∈ limn sk(Agn ) = 0 or limn sk(Agn ) = 0. ±1 Ag is stable if and only if W (A) and W (Ag) are invertible. • Proof. Let all snapshots of Ag be Fredholm, pass to a subsequence Ah of Ag which is regularly T h -Fredholm, by Proposition 2.28, and apply Theorem 2.26 to prove the ﬁrst assertion. The secondJ one easily follows from Theorem 2.9, and the last one again from Theorem 2.26 and Proposition 2.28.

For the whole sequence A Alp we further conclude from Theorem 2.26, Proposition 2.28 and Corollary 2.23 ∈ F

Theorem 2.32. Let A = An Alp . Then, A is a Fredholm sequence if and only if W (A) ±1 { } ∈ F and all operators W (Ah) with h A are Fredholm. In this case the α- and β-number of A equal ∈ H

+1 −1 α(A) = dim ker W (A) + max dim ker W (Ah) + dim ker W (Ah) , h∈H A (2.11) +1 −1 β(A) = dim coker W (A) + max dim coker W (Ah) + dim coker W (Ah) . ∈H h A A sequence is stable if and only if all snapshots and ±1 with are A Alp W (A) W (Ah) h A invertible. ∈ F ∈ H

Notice that every -Fredholm operator is Fredholm if dim X < . Thus, the last theorem, P ∞ together with Corollary 2.30 reveal that the Fredholm property of W (A) is already suﬃcient for the Fredholm property of A in this case. The stability criterion of Theorem 2.32 often appears in a slightly diﬀerent form in the literature and we want to close this paragraph with an outline of this alternative notation. For A = An { } in A p denote the set of all operators B which are -strong limits of one of the sequences F l h P

(V− [(I L ) + L A L ]V ) with h (or −) hn − hn hn hn hn hn ∈ H+ H + − + − by σstab(A) (or σstab(A), respectively). Of course, if Bh is in σstab(A) or σstab(A) then we can ±1 pass to a subsequence g of h and identify Bh with the respective operator W (Ag). ∈ HA

Theorem 2.33. A sequence A Alp is stable if and only if all operators in ∈ F + − σstab(A) := W (A) σ (A) σ (A) { } ∪ stab ∪ stab are invertible. 58 PART 2. FREDHOLM SEQUENCES AND APPROXIMATION NUMBERS

Remark 2.34. This result was recently proved in [43] for the ﬁnite section sequence (LnALn) of a single band-dominated operator A, also by studying its subsequences. It has a lot of predecessors, which required additional restrictions like p = 2, 1 < p < , a uniform bound for the norms of ∞ the inverses of all operators in σstab LnALn , or the existence of a predual setting (see [61], [64], [45], [44], [15], for example). { }

Now having this result for sequences in the whole algebra Alp , we get much more flexibility in constructing efficient algorithms for specific operators. Assume,F for example, that the operator A admits a decomposition m k

A = Aij i=1 j=1 X Y into band-dominated operators of simple structure (e.g. banded, triangular, or Toeplitz operators). This structure, which could permit the application of fast algorithms, gets lost if one applies the usual ﬁnite section method LnALn, but it can be preserved, if one uses the composition of the single ﬁnite sections LnAijLn instead. The results above provide the desired information on the stability and convergence also for such compositions.

2.5.2 Adapted finite sections In the preceding section the aim was to check if for arbitrarily given band-dominated operators A the “standard” finite section method, based on the projections Pn, applies. Providing that A is rich, the answer is that one has to check the Fredholm properties and the invertibility of a possibly infinite set σstab(A) of operators. Of course, we have seen that we can pass to subsequences to reduce the number of limit operators, but since the projections Ln are always chosen symmetric w.r.t. Z, the limiting processes towards and are somehow coupled, which seems to be artificial. ∞ −∞ Now we let A lp be fixed and we ask if there is a more adapted sequence of projections which provides∈ a specific A “finite section like” method for this single operator, such that we only have to check one snapshot at and one snapshot at and such that both can be chosen ∞ −∞ independently from each other. The idea is very natural and simple: Suppose that for A p ∈ Al (not necessarily rich) there is a sequence l −, such that A exists and, independently of l, ∈ H l let u + be another sequence such that Au exists. We show that the properties of the finite ∈ H l,u l,u sections Ln ALn , with { } l,u Ln = χ{l(n),...,u(n)}I, are determined by the three operators A, Al and Au.

Theorem 2.35. Let A lp , and let l −, u + be such that Al,Au σop(A) exist. l,u ∈ A ∈ H ∈ H l,u l,u l,u∈ l,u Further let A := An denote the sequence of the operators An := Ln ALn (im Ln ). Then { } ∈ L

+ − The operators A := χZ+ AlχZ+ I + (1 χZ+ )I and A := χZ AuχZ I + (1 χZ )I are • -Fredholm, whenever A is -Fredholm.− − − − − P P ± l,u + − If A and A are Fredholm then limn ind An exists, equals ind A + ind A + ind A and • l,u the approximation numbers from the right/left of An have the α-/β-splitting property with α = dim ker A + dim ker A+ + dim ker A−, and β w.r.t. cokernels instead of kernels.

± r l,u If one of the operators A, A is not Fredholm then, for each k N, limn sk(An ) = 0 or • l l,u ∈ limn sk(An ) = 0. A is stable if and only if A and A± are invertible. • 2.5. EXAMPLE: FINITE SECTIONS OF BAND-DOMINATED OPERATORS 59

l,u Proof. Notice that (Ln ) converges -strongly to the identity and introduce T := 1, 0, +1 , t l,u,t P l,u l,u,T l,u,T {− } homomorphisms En : (im Ln ) (En ) and sequence algebras , in the same way as in Section 2.5.1L. Then the→ finite L section sequence of each -regularizerF J of A is again in P l,u,T , is a l,u,T -regularizer for A (cf. (2.9)) and Theorems 2.6, 2.13, 2.14, 2.9, and 2.17 give Fthe claim. J Of course, these two slightly different approaches can be combined to get more appropriate methods for classes of band-dominated operators having a certain common structure: First, one l,u chooses the sequence (Ln ) of projections which align with the common structure in a sense and one then proceeds as in Section 2.5.1 to prove Theorems 2.31 and 2.32 also in this setting. Furthermore, [79] and [80] present an idea how one can pass to modified finite sections which need not to be stable, but which are still generalized invertible (e.g. Moore-Penrose invertible). Also note that the equations Ax = y and VκAx = Vκy are equivalent for every κ Z since Vκ is invertible, whereas at most one of them leads to a stable finite section sequence and∈ therefore can be solved by the finite section method. The simplest example of an operator for which such a preconditioning procedure is indicated to get a stable finite section sequence is the operator A = V1 itself. This method is known as index cancellation and was already studied in [24]. A comprehensive discussion can also be found in [36].

A final remark The core of the simplifications which appear in this more specific situation of finite sections of band-dominated operators is given by Proposition 2.28: The Fredholm properties of a sequence A are already determined by the Fredholm properties of the central snapshot T W (A), and hence the “ . -Fredholmness” can be extinguished from all results. This may look nice and simple for the moment,J but it is still unsatisfactory in a sense, because it can hardly be translated to other classes of operators or lifted to a more abstract level. Therefore, the next part is devoted to this gap, and the main tools there are localization techniques. This deeper study will provide further amazing relations between the sequences and their snapshots, for instance on the asymptotics of condition numbers and pseudospectra. 60 PART 2. FREDHOLM SEQUENCES AND APPROXIMATION NUMBERS Part 3

Banach algebras and local principles

In the previous part we have seen how the stability and invertibility problem in certain sequence algebras can be replaced by the invertibility problem for a family of operators (the snapshots) and for a coset in a smaller quotient algebra (namely T / T ). Of course, the latter condition is still unpleasant and can hardly be checked directly.F ThereforeJ it would be desirable to get a family of snapshots which is suﬃcient in the sense that the Fredholm property or invertibility of the snapshots can already provide the invertibility of the coset in question. It is not clear if or how this could be achieved in the very general context T . So, here we draw on well known local principles to attackF this question for classes of sequences which will be referred to as localizable sequences in what follows. For this we start with a short introduction and description of the required tools in Section 3.1 and proceed with their application to sequence algebras in 3.2. We achieve our goal, the characterization of the Fredholm property of localizable sequences solely in terms of its snapshots, with Theorem 3.9. Furthermore, we consider the asymptotic behavior of norms, condition numbers and pseudospectra. This approach was already evolved and published for the ﬁnite sections of band-dominated operators in [82], but is new in this abstract framework which permits to consider further applications in Part4. The third section is devoted to the pending proofs in the theory on approximate projections. Here we look at this concept from a new, more general and purely Banach algebraic point of view. 62 PART 3. BANACH ALGEBRAS AND LOCAL PRINCIPLES

3.1 Central subalgebras, localization and the KMS property

Classical Gelfand theory 1 A mapping ∗ : A A∗ on a Banach algebra is referred to as an 7→ A involution if for all A, B and all scalars α, β C ∈ A ∈ (A∗)∗ = A, (αA + βB)∗ =αA ¯ ∗ + βB¯ ∗ and (AB)∗ = B∗A∗.

We say that the unital Banach algebra with the involution ∗ is a C∗-algebra if for every A the equality A∗A = A 2 holds true.A Note that involutions on C∗-algebras are isometries,∈ A that is A∗ k= Ak fork everyk A . A Further,k a Banachk k k algebra is said∈ A to be commutative, if AB = BA holds for all A, B . C ∈ C So, in what follows let be a commutative C∗-algebra. A character of is a non-zero homomor- C C phism from into the algebra C. We denote the set of all characters by C and mention that every characterC is unital with norm 1. Every element A deﬁnes a complex-valuedM function ∈ C Aˆ on C, its so-called Gelfand transform, by the rule Aˆ(h) := h(A). The coarsest topology on M C which makes every Aˆ continuous is referred to as the Gelfand topology. AnM ideal is said to be maximal, if the only ideal of which is a proper superset of is itself. I ⊂ C C I C

Theorem 3.1. (Gelfand-Naimark Theorem) Let be a commutative C∗-algebra. Then the mapping h ker h is a bijection from the set C 7→ C of the characters h of onto the set of the maximal ideals of . Provided with the Gelfand M C C topology, C becomes a compact Hausdorﬀ space and the Gelfand transformation A Aˆ is an M 7→ isometric isomorphism from onto C( C), the algebra of all continuous functions on C. C M M Thus, one can think of the elements h of C either as characters on or as maximal ideals of . In the literature both characterizationsM are often used synonymously,C and one usually refers C to C as the maximal ideal space of . M C Due to this theorem, can be isometrically identiﬁed with the algebra of the continuous functions C on the maximal ideal space C. Clearly, an element A is invertible if and only if its Gelfand M ∈ C transform Aˆ does not have any zeros on C (one may say that it is locally invertible in every point). Furthermore the norm of an elementM A equals the norm of its Gelfand transform Aˆ. Since the latter is a continuous function on the compact set C, we have M A = max Aˆ(h) . (3.1) k k h∈M | | C

Therefore, in all what follows, the elements of C are denoted by x and are called points, the elements of are denoted by ϕ and are calledM functions, as a rule, and we will be a bit loosely and use theC notation ϕ(x) for ϕˆ(x). For the identiﬁcation of the maximal ideal space C one often beneﬁts from the Banach-Stone theorem, whose proof can be found in [21], V.8.8.M

Theorem 3.2. Let H, K be compact Hausdorﬀ spaces, and let T be an isometric isomorphism between C(H) and C(K). Then H and K are homeomorphic, that is there is a continuous bijection τ : H K whose inverse is continuous as well. → In the case of non-commutative Banach algebras the situation is much more involved and we here mainly build upon the paper [10] by Böttcher, Krupnik and Silbermann.

1These results and their proofs can be found in every textbook on C∗-algebras (e.g. [30], Section 4.1). 3.1. CENTRAL SUBALGEBRAS, LOCALIZATION AND THE KMS PROPERTY 63

Local principles for non-commutative Banach algebras Let be a unital Banach algebra. The center cen of is the set of all elements C with theA property that CA = AC for all A . Let Abe a CA∗-subalgebra of cen containing∈ A the identity. ∈ A C A For each point x in the maximal ideal space C we introduce x, the smallest closed ideal in containing x,2 and let φ denote the canonicalM mapping from J to / . In case = theA x A A Jx Jx A algebra / x consists of only one element Θ which is zero and the identity at the same time, and thereforeA J we may say that it is invertible and has norm 0. Theorem 3.3. (Allan [1], Douglas [19], see also [12], Theorem 1.35 or [63], Theorem 2.3.16f) The element A is invertible if and only if φ (A) is invertible in / for every x C. ∈ A x A Jx ∈ M The mapping C R+, x φx(A) is upper semi-continuous. If φx(A) is invertible then M → 7→ k k there is a neighborhood U of x in C such that φ (A) is invertible in / for every y U. M y A Jy ∈ This again provides a “local characterization” of invertibility in . We are also interested in a formula which represents the norm of an element A in terms ofA local norms as in (3.1). Notice that the set of all finite sums ϕ A with some ϕ , ϕ (x) = 0 and A forms a dense i i i i ∈ C i i ∈ A subset of x, and, by Proposition 5.1 of [10], J P φ (A) = inf ϕA : ϕ , 0 ϕ 1, ϕ 1 in a neighborhood of x . (3.2) k x k {k k ∈ C ≤ ≤ ≡ } Definition 3.4. Let be a unital Banach algebra and a central C∗-subalgebra which contains the identity. The algebraA is a KMS-algebra with respectC to 3 if for every A and ϕ, ψ with disjoint supports (i.e.A ϕ(x)ψ(x) = 0 for every x) C ∈ A ∈ C (ϕ + ψ)A max ϕA , ψA . (3.3) k k ≤ {k k k k} Theorem 3.5. (Böttcher, Krupnik, Silbermann, [10], Theorem 5.3) The algebra is a KMS-algebra with respect to if and only if for every A A C ∈ A A = max φx(A) . (3.4) k k x∈M k k C A look into the proof of this Theorem in [10] reveals that one only has to check for every single A (separately) that (3.3) holds for all ϕ, ψ with disjoint supports if and only if (3.4) is true. Thus, we can flexibilize the notion of∈ KMS-algebras C a little bit without any change in Theorem 3.5: Definition 3.6. Let be a unital Banach algebra and a central C∗-subalgebra containing the identity. A subalgebraB is said to be a KMS-algebraC with respect to if (3.3) holds for every A and ϕ, ψ A ⊂with B disjoint supports. C ∈ A ∈ C The aim of the following considerations is to use this concept of localization to find practicable criteria for the T -Fredholm property of a structured operator sequence in T as introduced in Part2. To be aJ bit more precise, we want to characterize a class of settingsF T where the t- Fredholm property of the snapshots implies “local T -Fredholmness” of the sequenceF and henceP its T -Fredholmness in the following sense: The localJ principle of Allan and Douglas suggests to lookJ for a suitable C∗-subalgebra and to replace the T -Fredholmness by the invertibility of all respective local cosets. As a secondC step one then mightJ try to identify these local cosets with the snapshots to get the desired result. Both the existence of and the identification are not achievable in the very general context of sequence algebras T , butC under additional conditions. In what follows we try to give a framework which is sufficientlyF restrictive to manage these two problems, but on the other hand, also sufficiently general to cover the algebras of more concrete operator sequences which we have in mind and which we take into the focus in the4th part of this thesis. 2 More precisely, Jx is the smallest closed ideal in A containing the kernel of the character x : C → C. 3In recent literature (e.g. [72]) such pairs (A, C) are also called faithful localizing pairs. 64 PART 3. BANACH ALGEBRAS AND LOCAL PRINCIPLES

3.2 Localization in sequence algebras

Let T be a sequence algebra as in Section 2.1 et seq. F 3.2.1 Localizable sequences

A localizing setting Let T be a closed algebra containing I such that every snapshot W t(ϕ) of every sequence ϕ C ⊂is F just a multiple ϕ(t)It of the respective identity, and such that ∈ C for t1, t2 T , t1 = t2 there is always a ϕ with ϕ(t1) = ϕ(t2). Further suppose∈ 6 that / := ϕ + : ϕ∈ C can be provided6 with an involution which turns / into a commutativeC GC∗-algebra.{ G From∈ C} Theorem 3.1 we know that / can be identified CwithG the set of all continuous functions on its maximal ideal space .C NoticeG that, for every MC/G t T , the mapping / C which sends ϕ + to ϕ(t) is a character, that means T can be understood∈ as a subsetC G of → . G MC/G Let M be a set T M C/G such that the closure of C/G M is either empty or intersects M. For every x ⊂M T⊂we M also fix one “direction” t = t(Mx) T\ and say that it is associated to ∈ \ x t(x) x ∈ x x x. In this way the objects E := E , and analogously , En, Ln, etc., and consequently the snapshots W x are defined at every point x M. P ˜x∈ ˜x Finally, for each x M, fix one function F C( C/G) (that is F / ) which takes values ˜∈x ∈ M ∈ Cx G x ˜x in [0, 1] such that F 1 in a neighborhood of x, choose one sequence F = (Fn ) F , and set x −x x x ≡ ∈ ˜x (Sn) := (En (Fn )Ln). Now, we say that defines a localizing setting. The functions F may be regarded as “local filter” in x, respectively.C Having such a localizing setting we are going to single out a family of sequences A T which can be studied using localizationC with respect to , and whose local representatives∈ are F already C characterized by the snapshots W t(A).

Localizable sequences Suppose that defines a localizing setting. A sequence A T is said to be -localizable if, for every x MC, ∈ F C ∈ W x(A) is liftable, that is Ax := Ex(SxW x(A)Sx) = Fx Ex(LxW x(A)) Fx T , • { n n n } { n n } ∈ F W x(A) is almost commuting with Fx, that is [W x(A),Sx] 0 as n , • k n k → → ∞ A + and Ax + belong to the commutant of / in / , that means they commute • withG every coset ϕG+ , ϕ , C G F G G ∈ C A + and Ax + are locally equivalent in x, that is for every > 0 there is a function • G G ϕ + / being equal to 1 in a neighborhood of x such that ϕ(A Ax) + < . G ∈ C G k − Gk The set of all -localizable sequences shall be denoted by T = T ( ). C T L L C Finally, suppose that there is a sequence T = Tn of projections such that T + is locally equivalent to zero in every x M, and{ let} ∈T Ldenote the set of all “truncated”G sequences ∈ MC/G \ T of the form TAT + λ(I T) + J with A T , a complex number λ and a sequence J T . − ∈ L ∈ J Remark 3.7. Of course, it is possible (and even the rather common case) to choose M = MC/G and T = I. Then also T = T holds, as we obtain from the next proposition. In this case we say that defines a globallyT localizingL setting. OtherwiseC is said to define a partially localizing setting. Notice that this model will enable us C to consider sequence algebras which contain the constant sequences A = A of the operators of { } interest as well as the sequence T of projections which constitutes the “finite section method”, and to study the finite section sequences of the form TAT + (I T). For an application of this kind see Section 4.3. − 3.2. LOCALIZATION IN SEQUENCE ALGEBRAS 65

Proposition 3.8. The sets T and T are Banach algebras, fulﬁll T T and contain as well as T as closed ideals. T L T ⊂ L G J The set / T := ϕ + T : ϕ forms a commutative C∗-subalgebra of T / T being -isomorphicC J to / { and JC( )∈. C} F J ∗ C G MC/G Proof. Obviously, T is a closed linear subspace of T . For A, B T and x M we have that L F ∈ L ∈ W x(AB) is almost commuting with Fx, AB + belongs to the commutant of / , AxBx T and G C G ∈ F

x x x x x x x x x x x x x x x x x F (AB) F A B = E (S S W (AB)S S S W (A)S S W (B)S ) − { n n n n n − n n n n } ∈ G since W x(A) and W x(B) are almost commuting with Fx, hence Fx(AB)xFx T , too. For every ∈ F t T with W t(Fx) = 0 we have W t((AB)x) = 0, and for all other t we know that the snapshot ∈t x t −t x t t W (F ) is a non-zero multiple dI of the identity, hence (En ((AB)n)Ln) converges -strongly −t x x x t t P if and only if (En (Fn (AB)nFn )Ln) converges -strongly. The only if is trivial, and for the if part let K be t-compact. Then it holds that 4P P 2 −t x t −t x 2 x t −t x x x t d KEn ((AB)n)Ln =G KEn ((Fn ) (AB)n)Ln =G KEn (Fn (AB)nFn )Ln, as well as the dual equation with K multiplied from the right. This shows that (AB)x T , i.e. ∈ F W x(AB) is liftable. Moreover, (AB)x + belongs to the commutant of / since G C G x x x x x x (x) x x x x x ϕ(AB) = ϕ E (S W (AB)S ) =G E (ϕ S W (A)S W (B)L ) { n n n } { n n n n n } x x x x (x) x x x x x (x) x x =G E (S W (A)S ϕ W (B)L ) =G E (S W (A)ϕ W (B)S ) { n n n n n } { n n n n } (3.5) x x x (x) x x x x x x x x x (x) =G En(LnW (A)ϕn SnW (B)Sn) =G En(LnW (A)SnW (B)Snϕn ) { x x x x x } { } =G E (S W (AB)S ) ϕ = (AB) ϕ { n n n } for every ϕ . These equalities also give that AxBx + and (AB)x + are locally equivalent in x 5 The proof∈ C of T being a Banach algebra can nowG easily be completed.G L Furthermore, T clearly forms an algebra since T2 = T and T is a closed ideal in T , and a simple approximationT argument yields its closedness. The set J is a closed ideal in bothF algebras T and T , as well as T in T . To prove this for T in T , itG suﬃces to consider the generating T T J T J Lτ sequences A = (Eτ (Lτ K)) with some τ T and K (E , τ ). Its snapshots are x-compact, n n ∈ ∈ K P P respectively, and hence are liftable and almost commuting with Fx. The membership of A to the commutant of / follows from the fact that the snapshot W τ ( ) of every -sequence is C G · C a multiple of the identity. To prove the local equivalence of A to the respective lifting notice that A and Ax are locally equivalent (they even coincide) in every x M with t(x) = τ, since ∈ W x(A) = K in these cases. Let x M be such that t(x) = τ. Then W x(A) = 0. Choose a function ϕ which is equal to 1 in∈ a neighborhood of τ but6 vanishes in a neighborhood of x. ∈ C Then ϕA A and ϕA is locally equivalent to zero in x. Hence A is locally equivalent to zero in x as well.− Now∈ G the assertion easily follows for all T -sequences. Furthermore, the inclusions T T T are proved. J T ⊂ L ⊂ F Clearly, / T is a commutative subalgebra of T / T and it inherits its involution from / . It remainsC toJ show that ϕ + = ϕ + T alwaysF J holds. The estimate “ ” is obvious. AssumeC G k Gk k J k ≥ that it is even proper for one ϕ, which means that there is an > 0 and a sequence J T such ∈ J that ϕ + > ϕ + J + ϕ + J + + . Without loss of generality we can also assume that ϕ +k =Gk 1. Fixk x k ≥ ksuch thatGkd := (ϕ + )(x ) fulﬁlls d 1 and x either belongs k Gk 0 ∈ MC/G G 0 | | ≥ − 0 4 Write an =G bn if kan − bnk → 0 as n → ∞. 5 x simply choose ϕ such that ϕ(I − F ) ∈ G. 66 PART 3. BANACH ALGEBRAS AND LOCAL PRINCIPLES to T or to the interior of T . In the latter case we choose a function ψ + / equal MC/G \ G ∈ C G to 1 in a neighborhood of x0, equal to zero on T and of norm one. In case x0 T we set ψ := I. ∈ Then B + := dI ψ(ϕ + J) + belongs to T / and is invertible with its inverse given by G − G F G a Neumann series. Thus, all of its snapshots are invertible. In case x0 T this contradicts the x0 x0 x0 ∈ fact that W (B) is -compact. If x0 / T then ψJ , hence W (B) is zero, a contradiction as well. P ∈ ∈ G

Theorem 3.9. The algebras T , T / and T / T are inverse closed in , / and T / T , respectively. The same holds trueL L withG T replacedL J by T . F F G F J L T A sequence A T is T -Fredholm if and only if all of its snapshots are t-Fredholm. It is Fredholm if and∈ Tonly ifJ all of its snapshots are Fredholm. It is stable if andP only if all of its snapshots are invertible. Proof. Let Com( / ) / denote the commutant of / , that is the set of all elements in / which commute withC G every⊂ F G element in / . It is well knownC G that Com( / ) / is a closedF G subalgebra of / , which contains T /C Gby definition. Furthermore, recallC G the⊂ homomorphisms F G F G L G φx from Section 3.1 which act on Com( / ) and notice that two cosets A+ , B+ Com( / ) C G G G ∈ C G are locally equivalent in x M if and only if φx(A + ) = φx(B + ). Indeed, the only if follows ∈ G G from the fact that φx(ϕ + ) = φx(I + ) for every ϕ being equal to 1 in a neighborhood G G ∈ C of x. For the if part let φx(A B + ) = 0. Then A B + can be approximated by finite − G − G sums of the form i ϕiCi + with some ϕi , ϕi(x) = 0 and Ci as mentioned after Theorem 3.3, and the assertionG follows. The respective∈ C local homomorphisms∈ F on the commutant Com( / T ) TP/ T of / T shall be denoted by Φ . C J ⊂ F J C J x Let A T be T -Fredholm and B T one of its regularizers. We want to show that B T . ∈ L J ∈ F ∈ L Let x M and consider the snapshot B := W x(B) which is a x-regularizer for A := W x(A) ∈ x P by Theorem 2.6. We first prove that limn [B,Sn] = 0 and that B is liftable. We again write k k x a =G b if lim a b = 0, and since I AB, I BA are -compact we have n n n k n − nk − − P x x x x x x x x B(I S ) =G B(I S )AB =G BA(I S )B =G (I S )B. − n − n − n − n The snapshot W x(Bx) of the sequence Bx := Ex(SxBSx) at the point x obviously exists, { n n n } and for those t T with W t(Fx) = 0 the snapshots W t(Bx) exist as well and equal zero. Let ∈ t T t(x) be such that W t(Fx) (which is a multiple dIt of the identity) does not vanish, and ∈ \{ } t set At := W t(Ax) = W t Ex(SxASx) . For every K (E , t), { n n n } ∈ K P E−t(Ex(SxB(Sx)2ASx))K E−t(Ex(SxASx))K k n n n n n k n n n n −t x x x k k ≥ En (E (S BS )) k n n n k 1 E−t(Ex(SxBA(Sx)3))K + G , t x x 2 n n n n n ≥ M M sup Sn B k k n∈N k k k k −t x x x 2 x where Gn := En (En(SnB[A, (Sn) ]Sn))K tend to zero in the norm as n . Thus, letting n go to infinity, we find AtK c K with a constant c > 0 independent→ of ∞K, which shows that At is injective, due tok its kt ≥-dichotomy.k k Analogously, one checks that At is surjective, that is (At)−1 exists, and it belongsP to (Et, t) by Theorem 1.14. We even have that L P H := d4It E−t(Ex(SxBSx))E−t(Ex(SxASx))Lt n − n n n n n n n n n converges t-strongly to zero, hence, for every t-compact K, P P [E−t(Ex(SxBSx))Lt d4(At)−1]K k n n n n n − k = E−t(Ex(SxBSx))Lt [At E−t(Ex(SxASx))Lt ](At)−1K + H (At)−1K k n n n n n − n n n n n n k 3.2. LOCALIZATION IN SEQUENCE ALGEBRAS 67 tends to zero as n . The dual relation with K multiplied from the left-hand side holds as → ∞ well, thus the snapshot W t(Bx) exists and equals d4(At)−1. We conclude that B is liftable. In the same way as in the equations (3.5) and with the help of the fact B BAB (Ex, x) we − ∈ K P find that Bx + is in the commutant of / . By B BAB T and G C G − ∈ J

ϕBAB =G BAϕB =G BϕAB =G BABϕ we see that B + is in the commutant of / as well. Hence it remains to check that B and Bx are locally equivalent.G Since the coset C G

x T x T x T Φx(B B + ) = Φx(BA(B B ) + ) = Φx(B(AB AB ) + ) − J − J − J x x T T = Φx(B(AB A B ) + ) = Φx(B(I I) + ) − J − J x T T equals zero, we have φx(B B + ) φx( ) := φx(J + ): J . If x M T then T − G ∈ xJ { G ∈ J } ∈ \ T φx( ) = φx( ), hence φx(B + ) = φx(B + ). On the other hand, if x T , then φx( ) J x G Gx G x x ∈x x J equals φx( ), hence we find a -compact K such that φx(B B + En(LnKLn) + ) = 0. J x x x P x x x − x { } G Consequently, W (B B + En(LnKLn) ) = 0, and since W (B B ) = 0, it even holds K = 0. − { } x − x Therefore, we get again that φx(B + ) = φx(B + ). Thus, the local equivalence of B and B is G G proved for every x M, B T follows, and yields the inverse closedness of T / T in T / T . To show the respective∈ assertions∈ L for T and T / simply consult TheoremL 2.17J. F J L L G Now, let A T be T -Fredholm and B T be a regularizer. Let us show that B T . If ∈T T T J ∈ L ∈ T T = I then = and there is nothing to prove. Otherwise A =J T TAT+λ(I T) with λ = 0 T F − 6 and we can multiply the equations AB =J T I and BA =J T I with (I T) from the left (right, − −1 resp.) and we get λ(I T)B =J T λB(I T) =J T (I T). Hence, B =J T TB + λ (I T) and −1 − − − − B =J T BT+λ (I T). Putting the first of these equations into the second one we finally obtain −1− T B =J T TBT + λ (I T) that is B . Therefore the assertions on the inverse closedness of T , T / and T / −T easily follow∈ as T well. TheoremsT T G 2.6, 2.17T andJ Corollary 2.23 show that all snapshots of a ( T -Fredholm / Fredholm J / stable) sequence A T are ( t-Fredholm / Fredholm / invertible). For the converse let ∈ T P A T be such that all snapshots are t-Fredholm. Due to the local principle of Allan and ∈ T P T Douglas in Theorem 3.3 it suffices to check that Φx(A + ) is invertible having an inverse x T x T T T J Φx(C + ) with C + / for every x C/G, respectively. For x M we have J T J ∈x F JT ∈x M x ∈x that Φx(A + ) equals Φx(A + ), and since W (A) is -Fredholm each -regularizer J x J P P B belongs to (E , x) by Theorem 1.14. Its lifting Bx := Ex(SxBSx) belongs to T and L P { n n n } F Bx + is in the commutant of / as the above considerations show. Moreover, G C G x x T x x x x x x T I B A + = Ln E (S BS )E (S AS ) + − J { − n n n n n n } J = L Ex(SxBA(Sx)3) + T = L Ex((Sx)4) + T , { n − n n n } J { n − n n } J x x T x x T hence Φx(I B A + ) (as well as Φx(I A B + )) equals zero. This gives the local − J − J invertibility in x M. Now let x0 M belong to the closure of C/G M. From Theorem 3.3 ∈ ∈ M \ T we can derive that there is a point y0 C/G M in a neighborhood of x0 such that Φy0 (A+ ), T ∈ M \ J which equals Φy0 (λI + ), is invertible as well, and consequently λ = 0. Since λ is independent J T 6 T of y C/G M we get the invertibility of all cosets Φy(A+ ) and hence the -Fredholmness ∈ M \ J J of A. Now, let all snapshots of A T be Fredholm (or even invertible). Then each of them is t- ∈ T P Fredholm, respectively, and with the above we can conclude that A is regularly T -Fredholm. J Corollary 2.24 gives the Fredholm property of A, and in case of invertible snapshots we have splitting numbers α(A) = β(A) = 0, hence stability from Theorem 2.21. 68 PART 3. BANACH ALGEBRAS AND LOCAL PRINCIPLES

Remark 3.10. The advantages of the described setting are the following: Suppose that one is interested in a certain subalgebra T and one wants to eliminate the unwieldy condition on the T -Fredholm property in theA results⊂ F on stability, convergence of approximation numbers or the indexJ formula for the sequences in . Then Proposition 3.8 shows that one only needs to ﬁnd a suitable and to prove that the generatorsA of are localizable w.r.t. , to guarantee that T . In manyC applications this simpliﬁes mattersA essentially because theC algebras under considerationA ⊂ L are often spanned by only a few rather basic elements. We again refer to the application in Section 4.3 for a telling example. Furthermore, Theorem 3.9 obviates the need for checking the T -Fredholm property and, moreover, the stated inverse closedness provides control over the respectiveJ regularizers and inverses. The latter is a crucial aid in the next section where we deal with the convergence of condition numbers.

3.2.2 Convergence of norms and condition numbers Within this section suppose that deﬁnes a localizing setting in T such that, additionally, C F x T 6 φx(A + ) W (A) Ag + g holds for every A , g + and x M, •k G k ≤ k k ≤ k G k ∈ T ∈ H ∈ T / is a KMS-algebra with respect to / . •T G C G and say that deﬁnes a faithful localizing setting. C T −1 Proposition 3.11. Let A = An . The norms An and A converge and { } ∈ T k k k n k t −1 t −1 7 lim An = max W (A) , lim An = max (W (A)) . n→∞ k k t∈T k k n→∞ k k t∈T k k

Proof. At ﬁrst, notice that f : C/G R+, x φx(A + ) is upper semi-continuous on M → 7→ k G kT C/G by Theorem 3.3 and constant on C/G M since A . If C/G M is not empty thenM choose x M which also belongs toM the closure\ of ∈ T M andM conclude\ that 0 ∈ MC/G \

φx(A + ) φx0 (A + ) for all x C/G M. k G k ≤ k G k ∈ M \ With the formula (3.4) from Theorem 3.5 this yields an x M such that for every g 1 ∈ ∈ H+ t sup W (A) Ag + g = lim sup Agn lim sup An = A + t∈T k k ≤ k G k n→∞ k k ≤ n→∞ k k k Gk

= max φx(A + ) = max φx(A + ) = φx1 (A + ) x∈M / k G k x∈M k G k k G k C G x W 1 (A) . ≤ k k

t1 x1 Since there is a t1 T with W (A) = W (A) we get that limn An exists and is determined as the maximum of∈ the norms of all snapshots. k k Now we consider the “inverses”. Suppose that one snapshot of A is not invertible. Then, by Theorem 2.17, the sequence A and each of its subsequences cannot be stable, that is there is −1 −1 no bounded subsequence of An , hence limn An = . If, conversely, all snapshots are { } k k ∞ −1 invertible then Theorem 3.9 yields that A is stable, a sequence whose nth entry is An (if An is invertible) is a -regularizer for A in T and its snapshots are (W t(A))−1. Apply the above to this sequence toG prove the second assertedT equation.

6 For the deﬁnition of the local homomorphisms φx on Com(C/G) see Section 3.1. 7We again use the convention kB−1k = ∞ if B is not invertible. 3.2. LOCALIZATION IN SEQUENCE ALGEBRAS 69

T Corollary 3.12. Let A = An be stable. Then the condition numbers of the operators A which are deﬁned by cond({ A}) ∈ := T A A−1 converge and their limit is n n k nkk n k t t −1 lim cond(An) = max W (A) max (W (A)) . n→∞ t∈T k k · t∈T k k

Remark 3.13. Let us give some more concrete criteria for a faithful localizing setting.

1. The ﬁrst of the stated conditions in its deﬁnition is true if, for every t T , ∈ t −t t BPt = lim En = lim En = lim Sn = 1. n→∞ k k n→∞ k k n→∞ k k

2. For the second condition notice that T / belongs to the commutant of / in the unital Banach algebra / . In general isT not completelyG contained in T ,8 butC G the flexibilized Definition 3.6 ofF theG KMS propertyC permits to write the conditionT in this form. It is T particularly fulfilled if, for every An , Bn and all ϕ = ϕn , ψ = ψn with ϕ + = ψ + = 1 and ϕψ { ,} the{ following} ∈ F holds 9 { } { } ∈ C k Gk k Gk ∈ G

ϕnAnϕn + ψnBnψn max An , Bn for every n N. k k ≤ {k k k k} ∈

The ﬁrst part is straightforwardly proved. Indeed, for A T and x M ∈ T ∈ x x x x x x x φx(A + ) = φx(A + ) A + = lim sup En(SnW (A)Sn) W (A) k G k k G k ≤ k Gk n→∞ k k ≤ k k and Theorem 1.19 further yields for every g ∈ H+ x −x x −x x W (A) lim inf Egn (Agn )Sgn lim sup Egn Agn Sgn lim sup Agn . k k ≤ n→∞ k k ≤ n→∞ k kk kk k ≤ n→∞ k k

For the second part notice that for every > 0 there is an integer N such that, for every k N, ≥

ϕkAkϕk + ψkBkψk max lim sup An , lim sup Bn + . k k ≤ { n→∞ k k n→∞ k k}

Hence, for every A, B T and ϕ, ψ with ϕ + = ψ + = 1 and ϕψ , ∈ F ∈ C k Gk k Gk ∈ G ϕAϕ + ψBψ + max A + , B + . k Gk ≤ {k Gk k Gk} Now let C T and f, g be such that the closures of the supports of f + and g + are disjoint. Choose∈ T ϕ, ψ ∈such C that the functions ϕ + , ψ + are of norm one,G have disjointG support and f ϕf, g ∈ Cψg . Then G G − − ∈ G 2 2 (f + g)C + = ϕ fC + ψ gC + = ϕfCϕ + ψgCψ + max fC + , gC + . k Gk k Gk k Gk ≤ {k Gk k Gk} Since every pair of continuous functions on a compact set having disjoint support can be approximated by pairs of continuous functions having disjoint closed supports, we obtain the KMS property of T / with respect to / . T G C G 8Fix x ∈ int(M \ T ) and a function ϕ being equal to 1 in x and identically zero on T . Then ϕ + G and ϕx + G are not locally equivalent in x. 9This idea already appeared in [10]. 70 PART 3. BANACH ALGEBRAS AND LOCAL PRINCIPLES

3.2.3 Convergence of pseudospectra

We turn our attention again to the asymptotic behavior of the pseudospectra spN, An which have been introduced and studied in Section 1.4 and we now suppose an additional condition which we adopt from [27] and [84].

Deﬁnition 3.14. A Banach space X is called complex uniformly convex if for every > 0 there exists a δ > 0 such that

x, y X, y and x + ζy 1 for all ζ C, ζ 1 always implies x 1 δ. ∈ k k ≥ k k ≤ ∈ | | ≤ k k ≤ − As already mentioned in Remark 1.51 this simpliﬁes matters essentially. The precise result, taken from Shargorodsky [84], Theorem 2.6 and Corollary 2.7, reads as follows.

∗ Theorem 3.15. Let Ω be a connected open subset of C, let X or X be complex uniformly convex, and let A :Ω (X) be an analytic operator-valued function. Suppose that there exists → L 0 a λ0 Ω such that the derivative A (λ0) of A in λ0 is invertible. If A(λ) M for all λ Ω then ∈A(λ) < M for all λ Ω. k k ≤ ∈ k k ∈ We note that every Hilbert space is complex uniformly convex (see [17]). For a further class of Banach spaces which are complex uniformly convex (and which are highly important for the present text), we borrow the following again from [84], preliminary to Theorem 2.5.

Proposition 3.16. .

1. Let (S, Σ, µ) be an arbitrary measure space and let X = Lp(S, Σ, µ) with 1 p < denote the set of all (equivalence classes of) Lebesgue measurable functions f on≤ (S, Σ∞, µ) with the p-th power of f being Lebesgue integrable. Then X is complex uniformly convex. If X = L∞(S, Σ, µ) isk thek space of all Lebesgue measurable and essentially bounded functions then X∗ is complex uniformly convex.

2. If, additionally, µ(S) < then lp(Z,X) (or (lp(Z,X))∗, resp.) is complex uniformly ∞ convex. This particularly includes the cases lp(Z, CN ), N N, where CN is provided with the p-vector norm and the counting measure. ∈

Proof. For the proof of X (or X∗) being complex uniformly convex see the mentioned discussion in [84]. Let µ(S) < . Then X := lp(Z,X) can be identiﬁed with Lp(S,ˆ Σˆ, µˆ), where the measure space (S,ˆ Σˆ, µˆ) is given∞ by

ˆ ˆ S := Z S, Σ := k Ak : Ak Σ and µˆ k Ak := µ(Ak), × { } × ∈ { } × ( ∈ ) ∈ ! ∈ k[Z k[Z kXZ and the ﬁrst part applies.

Deﬁnition 3.17. Let M1,M2,... be a sequence of non-empty subsets of C. The uniform (partial) limiting set

u-lim Mn p-lim Mn n→∞ n→∞ of this sequence is the set of all z C that are (partial) limits of a sequence (zn) with zn Mn. ∈ ∈ These limiting sets are closed as shown with [30], Proposition 3.2. 3.2. LOCALIZATION IN SEQUENCE ALGEBRAS 71

Proposition 3.18. Let define a faithful localizing setting and further let all spaces Et or their C T duals be complex uniformly convex. For A = An and every N Z+, > 0 { } ∈ T ∈ t u-lim spN, An = p-lim spN, An = spN, W (A). n→∞ n→∞ ∈ t[T Chapter 3 in the book [13] contains a concise discussion of the development and application of -pseudospectra as well as the present proposition in this particular case N = 0. We particularly point out again that results similar to Theorem 3.15, which guarantee that the resolvent is not locally constant, have been in that business from the very beginning, see Böttchers pioneering work [6], for example. We further should mention [9] which has presented the convergence results for -pseudospectra of the finite sections of operators on Lp-spaces, 1 < p < , for the first time. The present proof of Proposition 3.18 is an adapted variant of [13], Theorem∞ 3.17.

N Proof. Suppose z sp W t(A), that is W t(A zI) and hence W t((A zI)2 ) are not invertible. ∈ − − −2N Then A zI is not stable and Proposition 3.11 yields that limn (An zLn) = which − k − k ∞ implies z spN, An for suﬃciently large n, and therefore z u-limn spN, An. ∈ t ∈ t t −2N −2N So, now suppose that A zI is stable, but z spN, W (A), i.e. (W (A) zI ) − ∈ k − k ≥ for one t T . Let U be an arbitrary open ball around z such that W t(A) yIt is invertible ∈ N N − for all y U. If (W t(A) yIt)−2 would be less than or equal to −2 for every y U then ∈ k − k N N ∈ Theorem 3.15 would imply that (W t(A) zIt)−2 < −2 , a contradiction. For this notice k N − k that the function y (W t(A) yIt)−2 is analytic on U and its derivative in z is invertible. 7→ − t t −2N −2N Hence there is a y U such that (W (A) yI ) > , that is we can ﬁnd a k0 such that ∈ k − k −2N t t −2N 1 (W (A) yI ) > for all k k0. k − k − k ≥

Because U was arbitrary we can choose a sequence (zm)m s.t., for m 1/, zm belongs to the t ≥ −2N (N, 1/m)-pseudospectrum of W (A) and zm z as m . Since limn (An zmLn) − t t −2N → → ∞ k − k exists and equals maxt∈T (W (A) zmI ) , due to Proposition 3.11, it is greater than or N k − k N N equal to ( 1/m)−2 . Consequently, for suﬃciently large n, (A z I)−2 −2 and thus − k n − m k ≥ zm spN, An. This shows that z = limm zm belongs to the closed set u-limn spN, An. ∈ N N Finally consider the case that (W t(A) zIt)−2 < −2 for all t T . Then k − k ∈ −2N t t −2N −2N lim (An zLn) = max (W (A) zI ) < , n→∞ k − k t∈T k − k

−2N −2N hence there are a δ > 0 and an n0 N such that (An zLn) δ for all n n0. If y z is suﬃciently small and n ∈n we then furtherk − get k ≤ − ≥ | − | ≥ 0 N N (A yL )−2 = ((A zL )(L + (z y)(A zL )−1))−2 k n − n k k n − n n − n − n k N N = (A zL )−2 (L + (z y)(A zL )−1)−2 k n − n n − n − n k −2N −2N (An zLn) δ k − k−1 2N − −1 2N ≤ (1 z y (An zLn) ) ≤ (1 z y (An zLn) ) −N | − |k − k − | − |k − k < −2 .

Thus, z does not belong to p-limn spN, An. Since u-limn spN, An p-limn spN, An this completes the proof. ⊂ 72 PART 3. BANACH ALGEBRAS AND LOCAL PRINCIPLES

Proposition 3.6 in [30] states that for compact sets Mn the limits u-limn Mn and p-limn Mn 10 coincide if and only if Mn converge w.r.t. the Hausdorff distance (to the same limiting set). Thus, we can reformulate the preceding proposition as follows. Corollary 3.19. Let define a faithful localizing setting and further let all spaces Et or their C T duals be complex uniformly convex. For a sequence A = An the (N, )-pseudospectra of { } ∈ T the elements An converge w.r.t. the Hausdorff distance to the union of the (N, )-pseudospectra of all snapshots W t(A). This now clarifies the observation that we made in the beginning of Section 1.4.4: In general, one cannot expect that for a given increasing sequence (Tn) of projections the (N, )-pseudospectra of the truncations TnATn of an operator A converge to spN, A. The reason is that A captures only one facet of the asymptotic behavior of (TnATn), and (at least in many cases) further snapshots are needed to restore all relevant details of this sequence. We finally pose the question what happens if the spaces Et are not known to be complex uniformly convex, and Theorem 3.15 is not available. A thorough look into the proof of Proposition 3.18 reveals that this additional condition is only needed to conclude z u-limn spN, An for suffi- N N ∈ ciently large n from (W t(A) zIt)−2 −2 for one t T . Without it, we can arrive at the k − k ≥ N ∈ N same conclusion if we suppose that (W t(A) zIt)−2 is strictly greater than −2 for one t. k − k T Therefore, the picture for general Banach spaces is as follows: For every A , every N Z+ and 0 < δ < ∈ T ∈

t t spN,δ W (A) u-lim spN, An p-lim spN, An spN, W (A). ⊂ n→∞ ⊂ n→∞ ⊂ ∈ ∈ t[T t[T Thus we can use Theorem 1.44 to carry forward Theorem 1.48:

Corollary 3.20. Suppose that the index set T of the algebraic framework T is ﬁnite and T F C deﬁnes a faithful localizing setting. Let A = An . For every 0 < α < < β there is an { } ∈ T N0 Z+ such that, for every N N0, ∈ ≥

t t Bα sp W (A) u-lim spN, An p-lim spN, An Bβ sp W (A) . ⊂ n→∞ ⊂ n→∞ ⊂ ∈ ! ∈ ! t[T t[T Remark 3.21. Such results on the convergence of norms, condition numbers and pseudospectra have been proved for various classes of operators, such as Wiener-Hopf operators [4], Toeplitz operators [13], [8], band-dominated operators on l2 [57], or convolution type operators on cones [50]. A systematic and abstract presentation for the C∗-algebra case is in [30]. The present text provides several extensions: It is in a more abstract and versatile axiomatic formulation, which covers more exotic situations such as l∞ spaces, in particular the class of band-dominated operators on all lp-spaces as we will see in Section 3.2.5. Moreover, it treats (N, )-pseudospectra, and it overcomes one of the most unpleasant hurdles in that business: the inverse closedness of the algebras under consideration is automatically given.

3.2.4 Rich sequences meet localization

Let us condense the previous results and apply them to rich sequences A. For this, again HA stands for the set of all strictly increasing sequences g of positive integers such that Ag is a T -structured subsequence of A. Further, we denote by T the set of all sequences A T TR ∈ R 10This observation is due to Hausdorﬀ and can be found in his book [35] as well. 3.2. LOCALIZATION IN SEQUENCE ALGEBRAS 73

which have the following property: Every T -structured subsequence Ag has a subsequence Ah and an algebra (that may depend on A and h) which deﬁnes a faithful localizing setting (in T C T ), such that Ah . Fh ∈ Th Theorem 3.22. A sequence A T is Fredholm, if and only if all its snapshots are Fredholm. In this case ∈ T R

t t α(A) = max dim ker W (Ag) , β(A) = max dim coker W (Ag) . g∈H g∈H A ∈ ! A ∈ ! Xt T Xt T T Furthermore, for every A = An , { } ∈ T R t −1 t −1 lim sup An = max max W (Ag) , lim sup An = max max (W (Ag)) , ∈H ∈ ∈H ∈ n→∞ k k g A t T k k n→∞ k k g A t T k k t −1 t −1 lim inf An = min max W (Ag) , lim inf An = min max (W (Ag)) , n→∞ ∈H ∈ n→∞ ∈H ∈ k k g A t T k k k k g A t T k k and if, additionally, all spaces Et or their duals are complex uniformly convex then

t p-lim spN, An = spN, W (Ag). n→∞ g∈H , t∈T [A A sequence A T is stable if and only if all its snapshots are invertible. In this case ∈ T R

t t −1 lim sup cond(An) = max max W (Ag) max (W (Ag)) , →∞ g∈H t∈T k k · t∈T k k n A t t −1 lim inf cond(An) = min max W (Ag) max (W (Ag)) . n→∞ g∈H t∈T k k · t∈T k k A T Proof. Let all snapshots of A be Fredholm. Every T -structured subsequence Ag of A ∈ T R T has a subsequence Ah which belongs to , thus Theorem 3.9 yields that Ah is Fredholm and Th regularly T -Fredholm. Now, we derive the Fredholm property of A and formulas for α(A) and Jh β(A) from Theorem 2.26. For the converse Corollary 2.23 can be employed. The stability can be tackled in the same way. Recall from Proposition 3.11 that, for every g and the respective localizable subsequence ∈ HA Ah, t t max W (Ag) = max W (Ah) = lim Ahn lim sup An . t∈T k k t∈T k k n→∞ k k ≤ n→∞ k k

Now, choose a sequence j + such that limn Ajn = lim supn An and pass to a subsequence ∈ H t k k k k g A of j to see maxt∈T W (Ag) = limn Ajn = lim supn An . For the other equations we proceed∈ H analogously, with thek aid ofk Propositionsk k3.11, 3.18 andk Corollaryk 3.12.

3.2.5 Example: ﬁnite sections of band-dominated operators

p Back in the l setting, we want to apply the latest tools to the sequences in the algebra Alp which is generated by the ﬁnite section sequences of all rich band-dominated operators. By doingF this, we recover and extend the observations of [63], Section 6.3,11 as well as those of Section 2.5 in the present text.

11See also [69] and the preprint [57]. 74 PART 3. BANACH ALGEBRAS AND LOCAL PRINCIPLES

γ γ 1 b1 1 bn

1 1 n n γn n − − − ϕ bγ ϕ ◦ n

Figure 3.1: Inﬂating a function ϕ C[ 1, 1] via the splines bγ . ∈ − n

Theorem 3.23. Let and . Then is T -Fredholm if and only if its A = An Alp g A Ag t t { } ∈ F ∈ H J snapshots W (Ag) are -Fredholm, respectively. We further have the limits P 1 −1 C1 := lim Agn = max W (A) , W (Ag) , W (Ag) , n→∞ k k {k k k k k k} −1 −1 1 −1 −1 −1 C2 := lim Ag = max (W (A)) , (W (Ag)) , (W (Ag)) , n→∞ k n k {k k k k k k}

hence limn cond(Agn ) = C1C2 in the case of a stable sequence. If, additionally, lp or (lp)∗ is complex uniformly convex then the (N, )-pseudospectra of the operators Agn converge w.r.t. the Hausdorﬀ distance to the union of the (N, )-pseudospectra of the snapshots.12

Proof. The outline of this proof is very simple and straightforward: We are going to introduce γ T γ an algebra which deﬁnes a faithful globally localizing setting in g and turns Ag into a - localizable sequence,C hence allows to apply the results on the convergenceF of norms, conditionC numbers and pseudospectra. First of all, we set i1 := 1 and choose a subsequence i = (ik) of g such that

(t) t (t) t 1 sup (A W (Ag))Lk , Lk(A W (Ag)) : t T, gn ik < (3.6) k gn − k k gn − k ∈ ≥ k n o (t) t t for every k 2. This is possible since Agn tends -strongly to W (Ag). Now, for k N, define ≥ γP ∈ γn := k/2 for all n ik, . . . , ik+1 1 , and let bn : R [ 1, 1] denote the continuous piecewise ∈ { γ − } γ → − linear splines given by bn( n) = 1 and bn( (n γn)) = 1/2 which are constant outside the interval [ n, n], respectively.± ± ± − ± − γ For every continuous function ϕ C[ 1, 1] we let ϕn stand for the restriction of the function γ ∈γ − ϕ bn to Z. Thus, the sequence ϕnLn consists of operators of multiplication by inflated copies ◦ { } γ of ϕ with a certain distortion (see Figure 3.1). With ϕnLn ϕ ∞ it straightforwardly follows that the set k{ }k ≤ k k γ := ϕγ L : ϕ C[ 1, 1] C {{ n n} ∈ − } T t γ t forms a commutative subalgebra of , where W ( ϕnLn ) = ϕ(t)I for every t and every ϕ. ∗ γ γ F { } γ By : ϕnLn ϕnLn an involution is given and the mapping ϕ ϕnLn + is a -homomorphism{ from} 7→C[ { 1, 1] }onto γ / . We show that this mapping is even7→ { isometric,} G and therefore∗ γ / proves to be a− unital commutativeC G C∗-algebra and its maximal ideal space can be identified CwithG the interval [ 1, 1], due to Theorem 3.2. For this, we note that − γ γ ϕ ∞ ϕ L F ϕ L + k k ≥ k{ n n}k ≥ k{ n n} GkF/G γ holds and, on the other hand, ϕ ∞ = sup ϕ(x) : x [ 1, 1] ϕ L + follows k k {| | ∈ − } ≤ k{ n n} GkF/G 12Without the complex uniform convexity Corollary 3.20 is still applicable. 3.2. LOCALIZATION IN SEQUENCE ALGEBRAS 75 from 13

γ ϕ(x) = ϕ(x)I = -limV−b(bγ ) 1(x)cϕnLnVb(bγ ) 1(x)c | | k k kPn→∞ n − n − k γ γ γ γ 1 γ 1 lim inf V−b(bn)− (x)cϕnLnVb(bn)− (x)c lim sup ϕnLn = ϕnLn + F/G. ≤ n→∞ k k ≤ n→∞ k k k{ } Gk

Set M := [ 1, 1]. To every x ( 1, 1) we associate the direction t(x) := 0 T , and for every − ∈ − ∈ x [ 1, 1] we ﬁx a function Fx C[ 1, 1] which equals 1 in a neighborhood of x, equals 0 in a neighborhood∈ − of every t T x∈ and− takes only real values between 0 and 1. Then γ deﬁnes a faithful globally localizing∈ setting.\{ } For this apply Remark 3.13 and notice that C

(ϕγ A ϕγ + ψγ B ψγ )a = max ϕγ A ϕγ a , ψγ B ψγ a max A , B a (3.7) k n n n n n n k {k n n n k k n n n k} ≤ {k nk k nk}k k holds for all a En, all An , Bn and all ϕ, ψ C[ 1, 1] of norm 1 with disjoint support in the case p =∈ . For 1{ p} <{ }we ∈ Freplace (3.7) by∈ − ∞ ≤ ∞ (ϕγ A ϕγ + ψγ B ψγ )a p = ϕγ A ϕγ a p + ψγ B ψγ a p k n n n n n n k k n n n k k n n n k ϕγ A p ϕγ a p + ψγ B p ψγ a p max A p, B p a p. ≤ k n nk k n k k n nk k n k ≤ {k nk k nk }k k T T It remains to show that Ag g = g . For this assume that we can show, for every B lp , that ∈ T L ∈ A γ γ γ lim sup [V−kBVk, ϕn] = 0 for every ϕnLn . (3.8) n→∞ k∈Z k k { } ∈ C γ Then Alp / is obviously contained in the closed subalgebra Com( / ) of / which yields F G γ C G x F G that Ag + g commutes with every element of g / g. Every snapshot W (Ag) is again band- dominatedG (or at least the compression of a band-dominatedC G operator to the space Ex) and, x due to the choice of the functions Fg it is always liftable. Furthermore, by (3.8), it almost x x γ commutes with Fg and Ag + g belongs to Com( g / g) as well. Moreover, for every x ( 1, 1), G C G x x ∈ − equation (3.8) provides that sequences of the form ((I Ln)BFn) and (FnB(I Ln)), with B − − x being band-dominated, tend to zero in the norm as n , hence Ag + g and Ag + g easily prove to be locally equivalent in x. In case x = 1 (or→ in ∞ an analogousG way also for xG = 1) we choose ϕ C[ 1, 1] equal to 1 in a neighborhood of x and identically zero on [ 1, 1/2]−(or , respectively).∈ − Then γ x by the construction of the − and (3.6), [ 1/2, 1] ϕgn Lgn (Ag Ag ) g (γn) − { } − ∈ G T which yields the local equivalence in x again. Thus Ag g , and Theorem 3.9, Proposition 3.11 and Corollaries 3.12, 3.19 apply and give the assertions∈ of L the theorem. So, let us check the relation (3.8) for band-dominated operators B. It is easily proved for every shift operator and every operator of multiplication, hence it is clear for band operators and follows for band-dominated ones by a simple approximation argument.

T Clearly, for every sequence A Alp , the proof of the previous theorem shows that A , hence Theorem 3.22 applies, recovers∈ F and completes Theorem 2.32. More precisely, for∈ T every R −1 A = An Alp , we have formulas for the lim supn and the lim infn of both An and An , in terms{ } of ∈ norms F of the snapshots and their inverses. In case of a stable sequence,k k thisk is truek for cond(An) as well. Moreover, if the space lp or its dual is complex uniformly convex then, also by Theorem 3.22,

t 14 p-lim spN, An = spN, W (Ag). n→∞ g∈H , t∈T [A 13 γ −1 γ With (bn) being the inverse of the bijective function bn :[−n, n] → [−1, 1], and b·c the ﬂoor function. 14For a related observation in case p = 2 see [63], Theorem 6.3.15. 76 PART 3. BANACH ALGEBRAS AND LOCAL PRINCIPLES

3.3 -algebras P This section is devoted to the theory of Banach algebras with approximate projections. In particular, we give the pending proofs for the results that we stated in Section 1.2.1.

3.3.1 Basic deﬁnitions

Deﬁnition 3.24. Let be a unital Banach algebra and let = (Pn)n∈N be a bounded sequence of elements withA the following properties: P ⊂ A

Pn = I for all n N, and Pn = 0 for large n, • 6 ∈ 6 For every m N there is an Nm N such that PnPm = PmPn = Pm if n Nm. • ∈ ∈ ≥ Then is referred to as an approximate projection. In all what follows we set Qn := I Pn and we furtherP write m n if P Q = Q P = 0 for all k m and all l n. − k l l k ≤ ≥ -compactness Let be an approximate projection on . An element K is called - P P A ∈ A P compact if KPn K and PnK K tend to zero as n . By ( , ) we denote the set of all -compactk elements− k andk by −( )kthe set of all elements→ ∞A AforP whichK AK and KA are -compactP whenever K is -compact.A P ∈ A P P Theorem 3.25. Let be an approximate projection. ( ) is a closed subalgebra of , it contains the identity, andP ( , ) is a proper closed idealA ofP ( ). An element A belongsA A P K A P ∈ A to ( ) if and only if, for every k N, A P ∈ P AQ 0 and Q AP 0 as n . (3.9) k k nk → k n kk → → ∞ Proof. The following proof is an adaption of [63], Proposition 1.1.8. The conditions (3.9) are clearly necessary for A ( ). Let, conversely, A satisfy (3.9) and let ∈ A P K ( , ). Given > 0, choose k such that QkK < , and choose N such that QnAPk < for∈ all AnP KN. Then k k k k ≥ Q AK Q A Q K + Q AP K < ( Q A + K ) k n k ≤ k n kk k k k n kkk k k n k k k for all n N. The other conditions can be checked similarly, thus A ( ). It is immediate≥ from the definition that ( , ), ( ) are subalgebras∈ A ofP and that ( , ) A P K A P A A P K forms a proper ideal in ( ). To check the closedness of ( , ) let (Kn) ( , ) converge to K in the norm, andA P consider A P K ⊂ A P K ∈ A Q K Q K + Q K K . k n k ≤ k n mk k nkk − mk The latter term becomes as small as desired, if m is large, and the first one tends to zero as n for every m. Analogously, we find that KQ 0 as n , hence K ( , ). → ∞ k nk → → ∞ ∈ A P K If A is the (norm) limit of (An) ( ) then AK, KA are the limits of (AnK), (KAn) ( , ) for every K ( , ), hence are⊂ A -compactP as well. Thus, A ( ). ⊂ A P K ∈ A P K P ∈ A P -Fredholmness Here are two possible ways to introduce “almost invertibility”, in analogy to theP Fredholm property in operator algebras. For A we may assume that there is a -regularizer B , that is I AB, I BA are • -compact.∈ A P ∈ A − − P For A ( ) we may assume that the coset A + ( , ) is invertible in the quotient • algebra∈ A( P)/ ( , ). A P K A P A P K The big question is, if these definitions coincide for A ( ). ∈ A P 3.3. -ALGEBRAS 77 P

3.3.2 Uniform approximate projections and the -Fredholm property P Definition 3.26. Given an algebra with an approximate projection = (P ), we set S := P A P n 1 1 and Sn := Pn Pn−1 for n > 1. Furthermore, for every finite subset U N, we define − ⊂ P := S . is called uniform if CP := sup P < , the supremum over all finite U k∈U k P k U k ∞ U N. ⊂ Two approximateP projections = (P ) and ˆ = (F ) on are said to be equivalent if for every P n P n A m N there is an n N such that ∈ ∈

PmFn = FnPm = Pm and FmPn = PnFm = Fm.

In case of equivalent approximate projections and ˆ, one easily checks that ( , ) = ( ˆ, ), hence also ( ) = ( ˆ). P P A P K A P K A P A P Theorem 3.27. Let be a uniform approximate projection on and A ( ). Then there P A ∈ A P is an equivalent uniform approximate projection ˆ = (F ) on with C CP such that P n A Pˆ ≤ [F ,A] = AF F A 0 as n . k n k k n − n k → → ∞ Proof. Successively choose integers 1 = j i j i j ... such that for every l 1 2 2 3 3 P AQ (2l+2l)−1 s i and Q AP (2l+5l)−1 t i . (3.10) k s jl k ≤ ∀ ≤ l k t jl k ≤ ∀ ≥ l+1 n n Then, for all k, n N set Uk := i2n+k−1 + 1, . . . , i2n+k and Vk := j2n+k−2 + 1, . . . , j2n+k , n ∈ { n } { } as well as U := 1, . . . , i n and V := 1, . . . , j n , and ﬁnd 0 { 2 } 0 { 2 } k+2 −1 k+2 −1 n n PU APj2n+k 2 (2 n) and PU AQj2n+k (2 n) . k k − k ≤ k k k ≤ Thus k+1 −1 1 PU n AQV n (2 n) , yielding PU n AQV n . (3.11) k k k k ≤ k k k k ≤ n ∈ kXZ+ For n N we set ∈ − n 1 k Fn := 1 PU n − n k kX=0 and deduce that FnFn+1 = Fn+1Fn = Fn as well as

n k 1 n 1 n n F = P n = kP n P n C , (3.12) n Un k Un k Un k P k k n − n − ≤ n − ≤ k=1 k=1 j=1 k=j X X X X

that is ˆ = (F ) is an approximate projection. Further, and ˆ are equivalent, since P n P P

Pj2n 1 Fn = FnPj2n 1 = Pj2n 1 and FnPj2n+n 1 = Pj2n+n 1 Fn = Fn. − − − − −

For bounded subsets U R we ﬁnally introduce the elements FU as in Deﬁnition 3.26 and easily check that they can be⊂ represented in the form

− N 1 k F = P U N Wk kX=0 78 PART 3. BANACH ALGEBRAS AND LOCAL PRINCIPLES

with certain disjoint ﬁnite sets Wk N, k = 0,...,N 1. By an estimate similar to (3.12) we ˆ ⊂ − deduce that is uniform with CPˆ CP . It remains to consider the commutators of A and Fn. As a start, noticeP that ≤

AF = P n AF + Q n AF n Uk n i2 +n n k=0 Xn n

n n n n = PU APV Fn + PU AQV Fn + Qi2n+n APj2n+n 1 Fn, k k k k − kX=0 kX=0 −1 where the last term is less than CP n in the norm and the middle term is not greater than −1 CP n as well, due to the above estimate (3.11). The ﬁrst one equals

n k n k PU n APV n Fn 1 I + 1 PU n APV n k k − − n − n k k kX=0 kX=0 n 1 n k = P n AP n (P n P n ) 1 P n AQ n + F A, Uk Vk Uk 1 Uk+1 Uk Vk n n − − − − n kX=0 kX=0 −1 where we redeﬁne P n = P n := 0. The second item is smaller than n in the norm, thus U 1 Un+1 −

n 1 2CP + 1 AF F A P n A(I Q n )(P n P n ) + n n Uk Vk Uk 1 Uk+1 k − k ≤ n − − − n kX=0 n − 1 1 2CP n + 2CP + 1 P n A(P n P n ) + . Uk Uk 1 Uk+1 ≤ n − − n k=0 X

Now we obviously have

n 2 n P n AP n P n AP n Uk Uk 1 Uk Uk 1 − ≤ − k=0 l=0 k=0 X X k≡l Xmod 3

and this can be further estimated by

2 n n n n P n A P n + P n A P n  Uk  Uj 1  Uk  Uj 1   − − − l=0 k=0 j=0 k=0 j=0,j=6 k X  k≡Xl(3)  ≡X  k≡Xl(3)  ≡X    j l(3)   j l(3)        

2 n n n n P n A P n + P n AQ n P n .  Uk  Uj 1  Uk Vk Uj 1  ≤ − − l=0 k=0 j=0 k=0 j=0,j=6 k X  k≡Xl(3)  ≡X  k≡Xl(3) ≡X   j l(3)  j l(3)     

We conclude

n 2 −1 −1 P n AP n C A C + n C = 3C ( A C + n ), Uk Uk 1 P P P P P − ≤ k k k k k=0 l=0 X X

3.3. -ALGEBRAS 79 P

and with a similar estimate for P n AP n we ﬁnally get the assertion by Uk Uk+1

P −1 −1 6CP ( A CP + n ) + 2CP n + 2CP + 1 AF F A k k . k n − n k ≤ n

Now, we can answer the question on the correct deﬁnition of -Fredholmness aﬃrmatively. P

Theorem 3.28. Let = (Pn) be a uniform approximate projection on and A ( ). Then the coset A + ( , )Pis invertible in the quotient algebra ( )/ ( , )Aif and only∈ A ifP there is a -regularizer AB P K for A. In particular, every -regularizerA P forAAP K ( ) (if it exists) belongs toP ( ), and (∈ A) is inverse closed in . P ∈ A P A P A P A Proof. Obviously, the invertibility of the coset implies the existence of a -regularizer B. P Conversely, let B be such that P := I BA, P 0 := I AB ( , ). Theorem 3.27 yields ∈ A − − ∈ A P K an equivalent uniform approximate projection ˆ = (Fn) such that [A, Fn] 0 as n . P k k → → ∞0 Thus, for Gn := I Fn, we also have [A, Gn] 0 as n . Notice that PGn and GnP tend to zero as n − , hence (writingk A kB → if A →B ∞ 0 as n k ) k k k → ∞ n ∼n n k n − nk → → ∞ BG BG AB BAG B G B. n ∼n n ∼n n ∼n n

Therefore [B,Gn] 0 as n , and we conclude, for every k, that FkBGn and GnBFk tend to zerok as n k →. Thus →B ∞ ( ˆ) = ( ) due to Theorem 3.25.k k k k → ∞ ∈ A P A P These results are completely new and clear up the open problem which has been in the background of the limit operator method [63], [44] for several years. They were recently published in [81] for the case of operator algebras (X, ). This consistent picture can now be condensed into the following deﬁnition. L P

Deﬁnition 3.29. We say that A ( ) is -Fredholm if there is a -regularizer B for A. ∈ A P P P ∈ A

3.3.3 -centralized elements P We shortly discuss a class of elements which may be regarded as analogues to band-dominated operators in operator algebras, in a sense.

Deﬁnition 3.30. Let = (P ) be a uniform approximate projection on the Banach algebra P n A such that Pn+1Pn = PnPn+1 = Pn for every n N. Say that A is -centralized if ∈ ∈ A P

lim sup PkAQk+m , Qk+mAPk : k N = 0. m→∞ {k k k k ∈ } Proposition 3.31. The set ( , ) of all -centralized elements is a closed unital subalgebra of ( ) and contains ( , A) asP aC closed ideal.P A P A P K If A ( , ) is -Fredholm then each of its -regularizers is -centralized as well. ∈ A P C P P P In particular, ( , ) is inverse closed in and ( ). A P C A A P This observation will be picked up in the applications part 4.4.6 in order to extend the results on the Fredholm property and the stability of the ﬁnite sections of band-dominated operators to a larger class of operators which will be called weakly band-dominated operators. 80 PART 3. BANACH ALGEBRAS AND LOCAL PRINCIPLES

Proof. Apply Theorem 3.25 to get ( , ) ( ), and straightforwardly check that ( , ) is even a closed linear subspace of (A)PcontainingC ⊂ A P ( , ) and the identity. The estimateA P C A P A P K P ABQ P A P BQ + P AQ BQ k k k+2mk ≤ k k kk k+m k+2mk k k k+mkk k+2mk and its dual for Qk+2mABPk obviates that ( , ) is an algebra. Let B be a k-regularizer fork A ( , ).A ByP TheoremC 3.28, B already belongs to ( ), and in fact∈ the A sameP arguments lead to the∈ A proofP C of B belonging to ( , ): Since A is -centralized,A P we can replace (3.10) in the proof of Theorem 3.27 by the strongerA P C conditions P

l+2 −1 l+5 −1 sup Ps+rAQjl+r (2 l) s il and sup Qt+rAPjl+r (2 l) t il+1 r∈Z+ k k ≤ ∀ ≤ r∈Z+ k k ≤ ∀ ≥ and, instead of constructing only one approximate projection ˆ = (F ), we now consider the P n family ˆr = (F r), r N, given by P n ∈ n−1 r k F := Pr + 1 PU n+r, n − n k kX=0 where U + r := u + r : u U . Then we again find that these ˆr are uniform approximate projections with { ∈and} r as . Moreover,P there is a sequence CPˆr CP supr∈N [Fn,A] 0 n (m ) of integers m ≤n such that k k → → ∞ n n ≥ r r r r r Pmn+rFn = FnPmn+r = Pmn+r and Pmn+1+rFn = FnPmn+1+r = Fn for all . Now, set r r and check that r if as it was n, r N Gn := I Fn supr∈N [Gn,B] 0 n done in the∈ proof of Theorem 3.28−. Thus, we find that for everyk fixed nk →and all m→ ∞m ≥ n+1 k k k sup PkBQk+m = sup PkBGnQk+m sup PkGn BQk+m + sup Pk [Gn,B] Qk+m , k∈N k k k∈N k k ≤ k∈N k kk k k∈N k kk kk k k where the first summand vanishes since PkGn = 0 and the second one becomes arbitrarily small as n becomes large. With a dual estimate for supk Qk+mBPk this provides B ( , ) and easily finishes the proof. k k ∈ A P C

3.3.4 -strong convergence P Let = (P ) be an approximate projection in the unital Banach algebra . A sequence (A ) P n A n ⊂ A converges -strongly to A if, for all K ( , ), both (An A)K and K(An A) tend to 0 asP n . In this∈ case A we write A ∈ AAP K-stronglyk or A −= -limk A .k − k → ∞ n → P P n n As for Proposition 1.16 we check that a bounded sequence (An) in converges -strongly to A if and only if (A A)P 0 and P (A A) 0 forA every ﬁxed PP . ∈ A k n − mk → k m n − k → m ∈ P Deﬁnition 3.32. By ( , ) we denote the set of all bounded sequences (An) , which possess a -strong limitF inA (P ). Furthermore, say that is an approximate identity⊂ if A there is a P15 A P P universal constant DP > 0 such that A DP supn An for every sequence (An) ( , ) and the respective -strong limit A. k k ≤ k k ∈ F A P P Remark 3.33. Notice that is an approximate identity on if and only if there is a universal P A constant D > 0 such that A D supm PmA for every A ( ). Indeed, the implication only if is clear with A := kP kA, ≤ and the ifk partk follows with (∈A A) P ( , ) from n n n ∈ F A P A D sup PmA = D sup lim PmAn D sup sup PmAn D sup Pm sup An . n∈ k k ≤ m∈N k k m∈N N k k ≤ m∈N n∈N k k ≤ m∈N k k n∈N k k 15 That is, it is independent of the sequence (An). 3.3. -ALGEBRAS 81 P

Theorem 3.34. Let be an approximate identity on the Banach algebra with identity I. P A

If (An) ( ) converges -strongly to A then (An) is bounded and A ( ), i.e. • (A ) ⊂( A, P). P ∈ A ∈ A P n ∈ F A P The -strong limit of every (A ) ( , ) is uniquely determined. • P n ∈ F A P Provided with the operations α(A ) + β(B ) := (αA + βB ), (A )(B ) := (A B ), and • n n n n n n n n the norm (An) := supn An , ( , ) becomes a Banach algebra with identity I := (I). k k k k F A P The mapping ( , ) ( ) which sends (An) to its limit A = -limn An is a unital algebra homomorphismF A P → and A P P

A DP lim inf An . (3.13) k k ≤ n→∞ k k

Every approximate projection ˆ which is equivalent to forms an approximate identity • and ( , ˆ) = ( , ) holdsP true. P F A P F A P Let be uniform. If (A ) ( , ) has an invertible limit A and (B ) is a bounded • P n ∈ F A P n ⊂ A sequence such that (AnBn), (BnAn) converge -strongly to the identity, then (Bn) belongs to ( , ) with the limit A−1. P F A P

Proof. For the first assertion let K ( , ). Then we have AnK,KAn ( , ) and further A K AK , KA KA 0. Since∈ A P (K , ) is closed, this implies that∈ AAKP andK KA belong k n − k k n − k → A P K to ( , ), i.e. A is in ( ). Assume that (A ) is unbounded. Set BP := sup P and A P K A P n n k nk m0 := 1, and successively choose mi−1 ki li pi mi for all i N as follows: Given 3 ∈ m − fix k m − and choose l k such that Q A 2DP BP i . This is possible since i 1 i i 1 i i k ki li k ≥ (An)n is unbounded and (PmAn)n is bounded, hence (QmAn)n is unbounded for every fixed m. Since is an approximate identity there is an n such that Qki Ali 2 P PnQki Ali by Remark 3.33P . Now choose p n, p l and m p such thatk k ≤ D k k i i i i i

Q A 2 P P P Q A 2 P P P Q A , k ki li k ≤ D k n pi ki li k ≤ D B k pi ki li k hence P Q A i3. The series ∞ n−2P converges absolutely, hence it deﬁnes an k pi ki li k ≥ n=1 mn element P in the Banach algebra . This element easily proves to be -compact. Thus, (PAn)n A P P converges to PA in the norm. Therefore (PAn)n is bounded, which contradicts

−2 Qkn Ppn PAln n Qkn Ppn Pmn Aln Qkn Ppn Aln n PAln k k k k = k2 k . k k ≥ Q P ≥ (BP + 1)BP n (BP + 1)BP ≥ (BP + 1)BP k kn pn k

The second assertion is quite obvious: If Pm(An A) and Pm(An B) tend to zero as n goes to infinity for every fixed m, then P (A k B) = 0−for everyk mk , hence−A kB = 0 by Remark 3.33. m − − The proof of ( , ) being a normed algebra is straightforward, and we only note that if (A ), (B ) areF boundedA P and converge -strongly to A, B ( ), respectively, then n n P ∈ A P K(A B AB) K(A A)B + KA(B B) 0 k n n − k ≤ k n − nk k n − k → for every K ( , ), as n , since A ( ) implies KA ( , ). The estimate (3.13) ∈ A P K → ∞ ∈ A P ∈ A P K follows from the definition, since we can replace (An) by one of its subsequences which realize the lim inf and then cancel arbitrarily but finitely many elements at the beginning of this subsequence. m m m Finally, let ((Cn ))m be a Cauchy sequence of sequences (Cn ) ( , ), where C shall denote m m ∈ F A P the -strong limit of (Cn ), respectively. For every n, (Cn )m converges in to an element Cn, P A m and the sequence (Cn) is uniformly bounded. Furthermore, (3.13) yields that (C )m is a Cauchy 82 PART 3. BANACH ALGEBRAS AND LOCAL PRINCIPLES

sequence with a limit C ( ). Now one easily checks that (Cn) converges -strongly to C, thus ( , ) is complete.∈ A P P For theF A fourthP assertion apply the fact that ( , ) = ( ˆ, ), as well as Remark 3.33 with A P K A P K APm AFk Pm and AFm APk Fm for fixed m and sufficiently large k. k k ≤ k kk k k k ≤ k kk k −1 Given the sequence (Bn) in the last assertion, we set B := A , find that B ( ) by The- orem 3.28, and consequently K(B B) = KB(A A )B + KB(A B I)∈tend A P to zero in n − − n n n n − the norm as n for every K ( , ). The same holds true for (Bn B)K and thus (B ) ( , )→with ∞ limit B. ∈ A P K − n ∈ F A P 3.3.5 A look into the crystal ball Notice that it would have been sufficient for this thesis to prove these theorems for the algebras (X, ) of operators in Part1. Nevertheless, one argument for the present Section 3.3 is given byL itsP simplicity and beauty. Working on this abstract level reveals that these results are purely Banach algebraic. There is no need for additional tools or notions coming from functional analysis or operator theory. Another reason is that there is a natural connection between the local principle described in Section 3.1 and the present -algebraic framework. This will not be elaborated here in more detail, as it is not importantP for the classes of operators we have in mind here. We only mention that there are further families of operators and respective discretization methods in the literature - such as finite sections of Toeplitz operators having piecewise continuous symbol (see [29], [74]), or Cauchy singular integral operators and the collocation method (see [37], [38]) - for which the situation is much more involved, since there one needs to consider the algebras “modulo” T instead of to achieve a sufficiently large center for the localization. Maybe the following obser-J vation canG give a new perspective onto these topics and can contribute some better understanding of inverse closedness: Let be a unital Banach algebra and be a commutative C∗-subalgebra, not necessarily in A C the center. We again think of the elements ϕ as continuous functions on C. Now, for a ∈ C x M fixed x C, we suppose that there is a sequence (ϕn) of functions of norm one, being equal to∈1 Min a neighborhood of x and having nested and contracting⊂ C supports in the sense that x x x ϕn+1ϕn = ϕn+1 and, for every ψ which is equal to one in a neighborhood of x, there is an n x x x ∈ C x x x such that ϕnψ = ϕn. Then := (Ln) with Ln := 1 ϕn is a uniform approximate projection on and an element A isP x-compact iff it is locally− equivalent to zero in x in the sense that theA norms ψA and Aψ∈ A areP small for functions ψ with ψ = ψ(x) = 1 and small support. k k k k k k Denote the set of such elements by x. x J ( ) is the subset of for which x gives an ideal. AFurthermore,P A ( Ax) is x-FredholmJ iff there is a B such that I, AB and BA are locally equivalent∈ in Ax.P In thisP case B automatically belongs∈ to A ( x), too. Moreover, for given A ( x) we can replace x by an approximate projection, consistingA P of functions in again, which∈ A asymptoticallyP commutesP with A. C Probably, these observations could lead to another abstract local principle with lower requirements on commutativity, in a sense. At the moment this is just a conjecture, but it is to be hoped that some future work will clear it up. Part 4

Applications

Here we reap the fruit of our labor. We particularly want to demonstrate that the algebraic framework of the previous parts unifies several approaches which have been considered in the past, recovers lots of former results for special classes of operators and permits to extend them. Basically, these extensions are made into three directions: 1. Include more exotic cases such as p = 1, which stayed out of reach in most applications until now, and obliterate differences to the∞ Hilbert space case and the case of reflexive Banach spaces with -strong convergence. 2. Cover larger classes of operators, mainly by the notion of rich sequences.∗ 3. State some new results, particularly on spectral approximation. Firstly, we pick up again the operators of Section 1.4.4, being quasi-diagonal with respect to a certain sequence (Ln) of projections, and we have a look at their finite sections with respect to (Ln). Section 4.2 is devoted to band-dominated operators on Lp-spaces and the whole bundle of results which we already know for their discrete analogues on lp. We particularly rediscover the index formula for locally compact operators of Rabinovich, Roch and Roe [59], [58] and extend it into several directions. The multi-dimensional convolution operators, which are subject of Section 4.3, have been intensively studied by Mascarenhas and Silbermann [48], [49], [50]. The present approach now permits to modify their proofs and to obtain the results for larger classes of such operators on a larger scale of spaces.

In the ﬁnal Section 4.4 we brieﬂy address some more concrete subclasses of lp and, moreover, we want to catch a glimpse of further relatives beyond the world of band-dominatedA operators. 84 PART 4. APPLICATIONS

4.1 Quasi-diagonal operators

Deﬁnition 4.1. Let X be a Banach space. Let further A (X) and assume that there is a ∈ L bounded sequence (Ln) of projections on X, such that

[A, L ] := AL L A 0 as n . (4.1) k n k k n − n k → → ∞ 1 Then A is said to be quasi-diagonal with respect to (Ln). Proposition 4.2. Let A (X) be a quasi-diagonal operator w.r.t. (L ). ∈ L n 1. If A is invertible and [A, L ] < ( L 2 A−1 )−1 then L AL is invertible in (im L ). k n k k nk k k n n L n 2 −1 −1 2. If A is invertible, [A, Ln] < ( Ln A ) , y X is given, x X is the unique solution to Ax = y,k and x k im Lk isk thek uniquek solution∈ to L AL x ∈= L y, then n ∈ n n n n n x x A−1 ( [A, L ] x + L y y ) . k − nk ≤ k k k n kk nk k n − k Proof. This proof is a slight modiﬁcation of the one given with [14], Proposition 4.2. The estimate

L AL A−1L L = L (AL L A)A−1L L 2 A−1 [A, L ] < 1 (4.2) k n n n − nk k n n − n nk ≤ k nk k kk n k −1 yields the invertibility of LnALnA Ln by a Neumann argument, hence the invertibility of LnALn from the right. With a similar estimate the invertibility from the left can be shown as well. The second assertion is also straightforward with xn = Lnxn and:

x x A−1 A(x x) = A−1 AL x L AL x + L y y k n − k ≤ k kk n − k k kk n n − n n n n − k A−1 ( (AL L A)L x + L y y ) . ≤ k k k n − n n nk k n − k

Corollary 4.3. Let A (X) be a quasi-diagonal operator w.r.t. (Ln) and let y X. If A is invertible and (I L ∈)y L 0 as n then (L AL ) is stable and the solutions∈ x of the k − n k → → ∞ n n n equations LnALnxn = Lny converge in the norm to the solution x of Ax = y.

2 −1 −1 Proof. It only remains to mention that for all n such that [A, Ln] < (2 Ln A ) we even get the estimate 1/2 at the right hand side of (4.2), and easilyk findk that (kL ALk k A−1kL )−1 2. k n n n k ≤ Therefore the right inverses of LnALn (and also the left inverses) are uniformly bounded. Remark 4.4. The notion of quasi-diagonality is due to Halmos [28] and was initially studied for bounded linear operators A on a separable Hilbert space X. There, an operator A (X) ∈ L was said to be quasi-diagonal, if there was a sequence (Ln) such that A satisfied (4.1), and the projections (Ln) were additionally supposed to be compact and orthogonal, and to converge strongly to the identity. In such a setting, Halmos observed that A is quasi-diagonal (in this stricter sense) if and only if A = B + K where B is a block diagonal operator with respect to some orthonormal basis of X and K is compact. This means that there exist compact and orthogonal projections Lñ tending strongly to I which asymptotically commute with A and even fulfill Lñ+1Lñ = LñLñ+1 = Lñ for every n. Berg [3] pointed out that every normal operator is quasi-diagonal in this stricter sense, hence also every self-adjoint operator. So, this class contains a lot of interesting examples like almost Mathieu operators, discretized Hamiltonians, difference operators originated from PDE’s with constant coefficients, certain weighted shifts, etc. We will not go into great detail here, but we

1This definition is slightly more general than Definition 1.52. 4.1. QUASI-DIAGONAL OPERATORS 85 refer to the paper of Brown [14] which extensively treats the Hilbert space case, including the discussion of convergence rates for certain classes of quasi-diagonal operators and the definition of some explicit algorithms for two special cases. That paper is heavily based on the results of the Standard Algebra approach in [30], which provides the machinery to describe stability and several asymptotic properties of operator sequences in a Hilbert space setting. Of course, the approach of the present text suggests the expansion to the Banach space case, and this is what we do in the following.

The preceding observations reveal that a sequence of projections (Ln) which meets the requirements given in the deﬁnition of a quasi-diagonal operator A are fairly good candidates for the constitution of appropriate ﬁnite sections LnALn. So, in what follows, we additionally assume that (L ) is a sequence of -compact projections tending -strongly to the identity, where n P P = (Pn) is a uniform approximate identity on X, which equips X with the -dichotomy. Our Paim is to apply the results of the general theory of Parts2 and3 and to improveP Corollary 4.3.

Proposition 4.5. The set QDP,(Ln) of all operators which are quasi-diagonal w.r.t. (Ln) is a Banach subalgebra of (X, ) and contains the ideal (X, ). If A QD is -Fredholm L P K P ∈ P,(Ln) P and B is a -regularizer for A then B QDP,(Ln). In particular, QDP,(Ln) is inverse closed in (X, ). P ∈ L P Proof. Notice that for every fixed m and arbitrary k (I P )AP (I P )L AP + (I P )(I L )AP k − n mk ≤ k − n k mk k − n − k mk (I P )L AP + (I P )A(I L )P + (I P )[A, I L ]P ≤ k − n k mk k − n − k mk k − n − k mk (I P )L A CP + (CP + 1) A (I L )P ≤ k − n kkk k k kk − k mk + (CP + 1)CP [A, L ] , k k k where the second and third term of the last estimate can be made arbitrarily small by choosing k large, and the first term tends to zero as n for every fixed k, hence (I Pn)APm 0 as n (and P A(I P ) 0 by an analogous→ ∞ estimate, too). Thus QDk − is a subsetk → → ∞ k m − n k → P,(Ln) of (X, ). One easily checks that QDP,(Ln) is a closed subalgebra. For K (X, ) we have KL P PKP 0 as l , and for all l ∈ K P k − l lk → → ∞ [P KP ,L ] = (I L )P KP P KP (I L ) P K ( (I L )P + P (I L ) ) k l l n k k − n l l − l l − n k ≤ k lkk k k − n lk k l − n k vanishes for n . Thus K QDP,(Ln). Let A have a -regularizer B. Then B (X, ) by Theorem 1.14 →and ∞ further I ∈AB, I BA (X, ), henceP ∈ L P − − ∈ K P BL L B = B(L A AL )B + BL (I AB) (I BA)L B n − n n − n n − − − n = B(L A AL )B + B(L I)(I AB) (I BA)(L I)B, n − n n − − − − n − which also tends to zero in the norm as n . This finishes the proof. → ∞ Remark 4.6. At this juncture we want to point out again that the underlying general theory of the Parts2 and3 and all proofs also work in the classical setting (without ). More precisely, the P subsequent theorems remain true, if (Ln) is a sequence of compact projections which converge -strongly to the identity. In this case there is no need for an approximate projection and we∗ can replace (X, ) by (X), (X, ) by (X), -compact by compact, -FredholmP by L P L K P K P P Fredholm, -strong convergence by -strong convergence and set BP := 1. P ∗ The finite section sequences (L AL ) of quasi-diagonal operators A obviously converge - n n P strongly to A. So, let QD be the smallest closed subalgebra of containing all sequences L AL with A QD F , where is the Banach algebra of all boundedF sequences A with { n n} ∈ P,(Ln) F { n} An (im Ln). Notice that W (A) := -limn AnLn exists for every sequence A = An QD. ∈ L P { } ∈ F 86 PART 4. APPLICATIONS

Theorem 4.7. A sequence A = An QD is Fredholm if and only if W (A) is Fredholm. { } ∈ F In this case α(A) = dim ker W (A) and β(A) = dim coker W (A) are splitting numbers and limn ind An = ind W (A). l/r If W (A) is not Fredholm then infinitely many approximation numbers sk (An) tend to zero as n goes to infinity. A sequence A QD is stable if and only if W (A) is invertible. ∈ F T Proof. Let T := 0 and let denote the Banach algebra of all sequences A = An { } F ⊂ F { } such that (AnLn) converges -strongly. Denote the limit again by W (A). Further, introduce the ideal T := L KL +P G : K (X, ), G 0 . Then T . J {{ n n} { n} ∈ K P k nk → } FQD ⊂ F The set := αI : α C is a commutative closed subalgebra of T , the respective snapshots C { ∈ } F are multiples of the identity, / turns into a C∗-algebra (provided with (αI + )∗ := αI + ) C G G G with T as maximal ideal space. With M = T and F0 := I it is clear that defines a globally localizing setting. C

For A := LnALn with A QDP,(Ln) we have that W (A) = A, hence this snapshot is liftable { } ∈ and almost commuting with F0. The coset A + as well as the lifting A0 + belong to the G G commutant of / , and it is also obvious that A = A0, i.e. they are locally equivalent in 0. Consequently, theC G whole algebra is contained in T = T by Proposition 3.8. FQD L T Now apply Theorem 3.9 to prove that A is T -Fredholm if and only if W (A) is -Fredholm, and to derive the asserted criteria for the FredholmJ property and the stability. TheoremsP 2.13, 2.14 and 2.9 provide the rest.

Theorem 4.8. Let A = An QD and assume that BP = limn Ln = 1. Then { } ∈ F k k −1 −1 lim An = W (A) , lim An = (W (A)) , hence lim cond(An) = cond(W (A)), n→∞ k k k k n→∞ k k k k n→∞ the latter only for stable A. For every 0 < α < < β there is an N0 Z+ such that ∈

Bα (sp W (A)) u-lim spN, An p-lim spN, An Bβ (sp W (A)) ⊂ n→∞ ⊂ n→∞ ⊂ for every N N . If, additionally, X or its dual is complex uniformly convex then, for every ≥ 0 > 0 and every N Z+, the (N, )-pseudospectra of the operators An converge w.r.t. the ∈ Hausdorff distance to the (N, )-pseudospectrum of W (A). Proof. Clearly, this follows from Proposition 3.11 and Corollaries 3.12, 3.19 and 3.20 if defines a faithful localizing setting, which is almost trivial if we take Remark 3.13 into account.C We only note that two functions on T have disjoint support iff at least one of them equals zero. Thus T / is a KMS-algebra w.r.t. / . T G C G 4.2. BAND-DOMINATED OPERATORS ON LP (R,Y ) 87

4.2 Band-dominated operators on Lp(R,Y )

Let Y be a Banach space. Here we consider the spaces Lp = Lp(R,Y ), with 1 p < , of all ≤ ∞ (equivalence classes of) Lebesgue measurable functions f : R Y for which the p-th power of ∞ ∞ → f Y is Lebesgue integrable. By L = L (R,Y ) we further denote the space of all (equivalence k k classes of) Lebesgue measurable and essentially bounded functions f : R Y . Note that the Banach space structure is given by pointwise addition and scalar multiplication→ as well as the usual Lebesgue integral norm (or the essential supremum norm in the case p = ). In the same p p ∞ vein, we also deﬁne the spaces L (R−,Y ) and L (R+,Y ) of functions over the half axes. Let 1 p and let Pn stand for the operator of multiplication by the characteristic function ≤ ≤ ∞ p of the interval [ n, n) acting on L . Then := (Pn) forms a uniform approximate identity and Lp has the -dichotomy.− To prove the latter,P we note that in case 1 p < the dual space P ≤ ∞ (Lp(R,Y ))∗ can be identiﬁed with Lq(R,Y ∗) where 1/p + 1/q = 1, i.e. ∗ is an approximate identity on (Lp)∗, and the space L∞ is complete w.r.t. -strong convergence.P Now Theorem 1.27 applies. P For α R we consider the operator ∈ U : Lp Lp, (U f)(t) := f(t α) α → α − p of shift by α. Let A (L , ) and h = (hn) R be a sequence with hn as n . The ∈ L P ⊂ | | → ∞ → ∞ operator Ah is called limit operator of A with respect to h if

U− AU A -strongly. hn hn → h P The set σop(A) of all limit operators of A is again called the operator spectrum of A. Finally, an operator A is called rich if every sequence h Z whose absolute values tend to infinity has ⊂ an infinite subsequence g such that Ag exists. One may ask why the definition of rich operators is exclusively based on sequences of integers. The reason is that it would be a much (unnecessarily) stronger condition if all sequences of real numbers shall have a subsequence which leads to a limit operator, and the “set of rich operators” would dramatically shrink. Lindner gave a comprehensive explanation of this phenomenon in [44], Section 3.4.13. Nevertheless, one can replace Z in this definition by any other “two-sided infinite” set of isolated points (cf. Section 4.2.2). Definition 4.9. An operator A (Lp, ) is said to be band-dominated if, for every ϕ BUC, ∈ L P ∈ lim sup [A, ϕt,rI] = 0. t→0 r∈R k k

Let p denote the set of all such operators. AL Notice that this definition is in line with the class of (discrete) band-dominated operators as the third item of Theorem 1.55 shows, and also the treatment of the finite sections is very similar. Nevertheless we want to present a standalone consideration here and avoid to play on the results in the discrete setting which are scattered all over the whole text. In the very end of this section we finish with a short discussion of the connections between the discrete and the continuous case.

4.2.1 Standard ﬁnite sections

Set L := P , further define E := im L and let A p denote the Banach algebra which is n n n n F L ⊂ F generated by all sequences LnALn with rich A Lp . Clearly, for every sequence A = An { } ∈ A { } in A p , the limit F L W (A) := -lim AnLn Pn→∞ 88 PART 4. APPLICATIONS exists. In this way, a first natural snapshot is defined and, as in the discrete case, there are two further points of interest, namely where the operators Ln perform the truncation. More precisely, for a given sequence h = (hn) of positive integers tending increasingly to infinity we 0 ±1 set Ehn := im Lhn , T := 1, 0, +1 , I := I, I := χR I and {− } ∓ E0 p 0 := L (R,Y ) Lhn := Lhn E0 : (im L0 ) (E ),B B hn L hn → L hn 7→ E±1 := im I±1 L±1 := U L U hn ∓hn hn ±hn ±1 ±1 E : (im L ) (Eh ),B U±h BU∓h hn L hn → L n 7→ n n t t t t for every n. By := (Ln) uniform approximate identities on E are given such that the E have the t-dichotomyP and the sequences (Lt ) converge t-strongly to the identities It on Et. As hn P T P usual, we let h denote the Banach algebra of all bounded sequences Ahn of bounded linear F t { t } t operators Ahn (Ehn ) for which there exist operators W Ahn (E , ) for each t T such that ∈ L { } ∈ L P ∈ −t t t t E (Ah )L W Ah -strongly as n . hn n hn → { n }P → ∞ Furthermore, introduce A as the set of all strictly increasing sequences h of positive integers + H − such that W (Ah) and W (Ah) exist. Of course, for A := LnALn with rich A Lp and a strictly increasing sequence h there is always a subsequence g{ of} h. Since the∈ set A T of all ∈ HA R rich sequences in is a closed algebra (see Theorem 2.26), we see that every A ALp is rich. F ∈ F Proposition 4.10. Let and . Then there is a subalgebra T which defines A ALp g A ∈ F T∈ H C ⊂ F a faithful globally localizing setting in g and which turns Ag into a -localizable sequence in T T F C . In other words: A p . Tg F L ⊂ T R 2 Proof. Firstly, let us prove that σop(B) Lp for every B Lp . Let C σop(B). From Theorem 1.19 we infer that C (Lp, ), but⊂ A we assume that∈ there A is a function∈ ψ C[ 1, 1], ∈ L P ∈ − a sequence (tn) R, tn > 0 and tending to zero, such that ∈ C := lim sup [C, ψt ,rI] = lim sup [UrCU−r, ψt I] > 0. n→∞ n n→∞ n r∈R k k r∈R k k

Clearly, we can choose a sequence (r ) such that C = lim [U CU− , ψ I] , and for all n n k rn rn tn k sufficiently large n we fix an integer m with C/2 < [U CU− , ψ I]P . Since C is a limit n k rn rn tn mn k operator of B, we can fix real numbers s such that C/4 < [U U− BU U− , ψ I]P for n k rn sn sn rn tn mn k large , hence for all large , contradicting p . n C/4 < supr∈R [UrBU−r, ψtn I] n B L Let g , set i := 1 and constructk a subsequencek i = (i ) of g such that ∈ A ∈ HA 1 k (t) t (t) t 1 sup (A W (Ag))Lk , Lk(A W (Ag)) : t T, gn ik < (4.3) {k gn − k k gn − k ∈ ≥ } k for every k 2. Now, for k N, define γn := k/2 for all n ik, . . . , ik+1 1 , and let γ ≥ ∈ ∈ { γ − } bn : R [ 1, 1] denote the continuous piecewise linear splines given by bn( n) = 1 and γ → − ± ± bn( (n γn)) = 1/2 which are constant outside the interval [ n, n], respectively. For every ± − ± γ γ − continuous function ϕ C[ 1, 1] we define ϕn := ϕ bn, a continuous function over R (see ∈ − γ ◦ also Figure 3.1 on page 74). Thus, the sequence ϕnLn consists of multiplication operators by { }γ inflated copies of ϕ with a certain distortion. Since ϕnLn = ϕ ∞ and with the involution ∗ : ϕγ L ϕγ L the set k{ }k k k { n n} 7→ { n n} γ := ϕγ L : ϕ C[ 1, 1] C {{ n n} ∈ − } 2Here, we proceed in the same manner as in the discrete case (Theorem 3.23). 4.2. BAND-DOMINATED OPERATORS ON LP (R,Y ) 89 becomes a commutative C∗-subalgebra of T being isometrically isomorphic to C[ 1, 1], and t γ t F γ − W ( ϕnLn ) = ϕ(t)I for every ϕ and t T . Moreover, the mapping ϕ ϕnLn + is an isometric{ }-isomorphism and therefore γ /∈ also proves to be a unital commutative7→ { }C∗-algebraG whose maximal∗ ideal space can be identifiedC G with the interval [ 1, 1]. For this, we note that for every m − γ γ γ ϕ ∞ = ϕmLm = lim sup ϕnLn = ϕnLn + F/G. k k k k n→∞ k k k{ } Gk Set M := [ 1, 1]. To every x ( 1, 1) we associate the direction t(x) := 0 T , and for every − ∈ − ∈ x [ 1, 1] we fix the function Fx C[ 1, 1] which equals 1 in a neighborhood of x, equals 0 in a neighborhood∈ − of every t T x ∈and− takes only real values between 0 and 1. Then γ defines a faithful globally localizing∈ setting.\{ } This is immediate from the definition and RemarkC 3.13 since, in the case p = , ∞ (ϕγ A ϕγ + ψγ B ψγ )f = max ϕγ A ϕγ f , ψγ B ψγ f max A , B f (4.4) k n n n n n n k {k n n n k k n n n k} ≤ {k nk k nk}k k holds for all f En, all An , Bn and all ϕ, ψ C[ 1, 1] of norm 1 with disjoint support. For 1 p < ∈we replace{ (4.4} {) by} ∈ F ∈ − ≤ ∞ (ϕγ A ϕγ + ψγ B ψγ )f p = ϕγ A ϕγ f p + ψγ B ψγ f p k n n n n n n k k n n n k k n n n k ϕγ A p ϕγ f p + ψγ B p ψγ f p ≤ k n nk k n k k n nk k n k max A p, B p f p. ≤ {k nk k nk }k k T T It remains to show that Ag belongs to = . For this assume that we can show, for every Tg Lg B Lp , that ∈ A γ γ γ lim sup [U−kBUk, ϕn] = 0 for every ϕnLn . (4.5) n→∞ k∈Z k k { } ∈ C γ Then ALp / is obviously contained in the closed subalgebra Com( / ) of / which yields F G γ C G x F G that Ag + g commutes with every element of g / g. Every snapshot W (Ag) is again band- dominatedG (or the compression of a band-dominatedC G operator to the space Ex) and, due to x the choice of the functions Fg it is always liftable. Furthermore, by (4.5), it “almost commutes x x γ with Fg ” and Ag + g belongs to Com( g / g) as well. Moreover, for every x ( 1, 1), the G C G x x ∈ − assumption (4.5) provides that sequences of the form ((I Ln)BFn) and (FnB(I Ln)), with − − x B being band-dominated, tend to zero in the norm as n , hence Ag + g and Ag + g easily prove to be locally equivalent in x. If x = 1 (or in→ an ∞ analogous way alsoG for x = G1) we choose ϕ C[ 1, 1] equal to 1 in a neighborhood of x and identically zero on [ 1, 1/2]−(or , respectively).∈ − Then γ x by the relation (4.3) which− yields the [ 1/2, 1] ϕgn Lgn (Ag Ag ) g local− equivalence in x again and{ finishes} the proof.− ∈ G γ So, let us prove the relation (4.5) for a fixed B Lp . Unfortunately, the ϕn do dot emerge from a linear inflation (as the functions in the Definition∈ A 4.9 of band-dominated operators do), γ since we used the piecewise linear splines bn. To tackle this, we simply decompose ϕ into three continuous functions, each of them being non-constant on at most one of the intervals [ 1, 1/2], γ − − [ 1/2, 1/2], [1/2, 1]. Then the bn inflate each of these functions linearly, with some additional shift− and with disparate speed, and the definition of band-dominated operators easily gives the claim. Indeed, if ϕ( 1/2) = 0 then we can split ϕ into the sum two functions ϕ1, ϕ˜ being supported only below− or above 1/2, respectively. Otherwise, if ϕ( 1/2) = 0, then we can write the function as the product of− two functions ϕ , ϕ˜, where ϕ is constant− 6 above 1/2 and ϕ˜ is 1 1 − constant below this critical point. Now do the same with the function ϕ˜ to split it into ϕ2, ϕ3 at the critical point 1/2. 90 PART 4. APPLICATIONS

Theorem 4.11. Let A = An ALp and g . { } ∈ F ∈ HA ±1 Ag is a Fredholm sequence if and only if W (A),W (Ag) are Fredholm. In this case • +1 −1 lim ind Agn = ind W (A) + ind W (Ag) + ind W (Ag) (4.6) n→∞

and the approximation numbers from the right/left of the entries of Ag have the α-/β- splitting property with +1 −1 α = dim ker W (A) + dim ker W (Ag) + dim ker W (Ag), +1 −1 β = dim coker W (A) + dim coker W (Ag) + dim coker W (Ag).

±1 r If one of the snapshots W (A) and W (Ag) is not Fredholm then limn sk(Agn ) = 0 or • l limn s (Agn ) = 0 for each k N. k ∈ ±1 Ag is stable if and only if W (A) and W (Ag) are invertible. • With the notation B−1 := if B is a non-invertible operator, we further have k k ∞ 1 −1 C1 := lim Agn = max W (A) , W (Ag) , W (Ag) , n→∞ k k {k k k k k k} −1 −1 1 −1 −1 −1 C2 := lim Ag = max (W (A)) , (W (Ag)) , (W (Ag)) , n→∞ k n k {k k k k k k} hence, for stable sequences Ag, even limn cond(Agn ) = C1C2 holds true. p p In the case L = L (R, C) the (N, )-pseudospectra of the operators Agn converge w.r.t. the Hausdorﬀ distance to the union of the (N, )-pseudospectra of the snapshots. T T Proof. Since Ag g , Theorem 3.9 tells us that Ag is g -Fredholm if and only if its snapshots are t-Fredholm,∈ itT is Fredholm if and only if all snapshotsJ are Fredholm, and it is stable if andP only if all snapshots are invertible. Theorems 2.13, 2.14, 2.9, Proposition 3.11 and Corollar- ies 3.12, 3.19 give the assertions.

For the whole sequences A ALp we apply Theorem 3.22 and Proposition 3.16 to get ∈ F

Theorem 4.12. A sequence A ALp is Fredholm, if and only if all of its snapshots are Fredholm. In this case ∈ F

t t α(A) = max dim ker W (Ag) , β(A) = max dim coker W (Ag) . g∈H g∈H A ∈ ! A ∈ ! Xt T Xt T

A sequence A ALp is stable if and only if all its snapshots are invertible. Furthermore, for ∈ F every A = (An) ALp , ∈ F t −1 t −1 lim sup An = max max W (Ag) , lim sup An = max max (W (Ag)) , ∈H ∈ ∈H ∈ n→∞ k k g A t T k k n→∞ k k g A t T k k t −1 t −1 lim inf An = min max W (Ag) , lim inf An = min max (W (Ag)) , n→∞ ∈H ∈ n→∞ ∈H ∈ k k g A t T k k k k g A t T k k and, for stable A, even

t t −1 lim sup cond(An) = max max W (Ag) max (W (Ag)) , →∞ g∈H t∈T k k · t∈T k k n A t t −1 lim inf cond(An) = min max W (Ag) max (W (Ag)) . n→∞ g∈H t∈T k k · t∈T k k A p p If L = L (R, C) and A ALp then ∈ F t p-lim spN, An = spN, W (Ag). n→∞ g∈H , t∈T [A 4.2. BAND-DOMINATED OPERATORS ON LP (R,Y ) 91

4.2.2 Adapted ﬁnite sections

Until now, we have ingenuously used Ln = Pn to get the simplest possible finite section method, but we had to pay the price that all operators under consideration needed to be rich. If, on the other hand, A is a (not necessarily rich) band-dominated operator and l, u are unbounded strictly decreasing or increasing sequences of negative/positive real numbers, respec- l,u tively, such that Al,Au σop(A) exist, then one can also set Ln = χ[l(n),u(n))I and consider ∈ l,u l,u the adapted finite section sequence A = Ln ALn . Then naturally, A is T -structured in the respective modified setting T , and its snapshots{ are} F

p and p A, χ−Au L (R ), χ+Al L (R+), | − | with χ−, χ+ standing for the characteristic functions of the half axes R− and R+, respectively. We again ﬁnd

A is Fredholm if and only if its three snapshots are Fredholm. In this case its α- and • β-numbers are determined as the sum of the kernel- (resp. cokernel-) dimensions of the snapshots. Furthermore

l,u l,u p p (4.7) lim ind Ln ALn = ind A + ind χ−Au L (R ) + ind χ+Al L (R+). n→∞ | − |

If one snapshot is not Fredholm then all approximation numbers of the ﬁnite sections from • at least one side tend to zero.

A is stable if and only if its three snapshots are invertible. • It holds that • l,u l,u p p lim Ln ALn = max A , χ−Au L (R ) , χ+Al L (R+) , n→∞ k k {k k k | − k k | k} as well as an analogous equation for the inverses. This also provides a formula for the limit of the condition numbers in case of a stable sequence.

If Lp = Lp(R, C) then the (N, )-pseudospectra of the ﬁnite sections converge to the union • of the (N, )-pseudospectra of the snapshots with respect to the Hausdorﬀ metric. p For general we set p p Then Corol- L (R,Y ) S := sp A sp(χ−Au L (R )) sp(χ+Al L (R+)). ∪ | − ∪ | lary 3.20 ensures at least, that for every 0 < α < < β there is an N0 Z+ such that l,u l,u l,u l,u ∈ Bα(S) u-lim spN,(Ln ALn ) p-lim spN,(Ln ALn ) Bβ(S) for every N N0. ⊂ n→∞ ⊂ n→∞ ⊂ ≥ In what follows we want to turn our attention to more concrete classes of band-dominated operators on Lp.

4.2.3 Locally compact operators In [58] Rabinovich and Roch studied band-dominated operators of the form A = I + K where K is locally compact. At this, a band-dominated operator K is said to be locally compact if ϕA and AϕI are compact operators for each function ϕ BUC with bounded support. ∈ The operators χ+Kχ−I and χ−Kχ+I are compact for each locally compact K (see [58]). Thus, p p the operators χ+A (L (R+)) and χ−A (L (R−)) are Fredholm operators, whenever A = I + K is Fredholm.∈ L We call ∈ L

ind+ A := ind(χ+A) and ind− A := ind(χ−A) 92 PART 4. APPLICATIONS

the plus- and the minus-index of A and ﬁnd that ind A = ind+ A + ind− A. Further notice that the limit operators of a locally compact operator are locally compact again. Then the main result of [58] for operators on the spaces Lp, 1 < p < reads as follows: ∞ Theorem 4.13. Let A = I + K with K being a rich locally compact operator. 1. The operator A is Fredholm iﬀ all limit operators of A are invertible and their inverses are uniformly bounded. 2. If A is Fredholm then, for arbitrary limit operators B,C σ(A) with respect to sequences ∈ l, u Z tending to ,(+ ), respectively, ⊂ −∞ ∞ ind+ A = ind+ B, ind− A = ind− C, hence ind A = ind+ B + ind− C.

This can also be obtained and even generalized applying the results of the present text. It is easy to check that for all band-dominated operators A = C + K with C invertible and K locally compact, and for all p [1, ] the -Fredholmness of A already implies its Fredholmness (see e.g. [44], Proposition 2.15).∈ ∞ Furthermore,P the -dichotomy of A also yields the converse implication. Consequently, for every p [1, ] andP every rich band-dominated operator of the form A = C + K we conclude from Theorem∈ ∞ 1.58 that A is Fredholm iff its limit operators are invertible and their inverses are uniformly bounded. If A = I +K is Fredholm then ind+ B and ind− B are finite numbers for every limit operator B of A and their sum equals zero since B is invertible. Moreover, the finite sections of locally compact operators are compact, hence the finite sections of A = I + K are Fredholm of index 0. Thus, Theorem 4.11 covers the second part of Theorem 4.13 and extends it to spaces Lp with p [1, ]. Moreover, formula (4.6) gives an extension to the finite section sequences of general rich∈ band-∞ dominated operators and even to sequences in the algebra ALp . By the analogon (4.7), also non-rich operators can be considered. A first version of such anF index formula for band-dominated operators in terms of limit operators was derived by Rabinovich, Roch and Roe in [59].

4.2.4 Convolution type operators We turn to another more concrete subclass of operators which were already studied by several authors. Thus, we omit repeating some details and refer the reader to e.g. [44], Section 4.2. With every function k L1 = L1(R, C), we associate the operator that maps a given function ∈ f Lp = Lp(R, C) to the so-called convolution k f which is given by ∈ ∗ (k f)(x) = k(x y)f(y)dy, x R. ∗ − ∈ ZR This operator is band-dominated and bounded by k 1, and it is usually denoted by C(a) where a, the symbol of C(a), is the Fourier transform ofk kk. Notice further that C(a) is always shift invariant, hence rich. Let p denote the Banach subalgebra which is generated by all such convolution operators and all richB operators bI of multiplication by a function b L∞. Moreover, introduce 0, the Banach ∈ Bp algebra generated by all operators of the form b1C(a)b2I where, again, a is the Fourier transform of an L1-function and b I, b I are rich multiplication operators with functions b , b L∞. 1 2 1 2 ∈ Proposition 4.14. (cf. [44], Lemma 4.10, Proposition 4.11) All operators in 0 are locally compact. • Bp The decomposition = bI : b L∞, bI rich 0 holds. • Bp { ∈ } ⊕ Bp 4.2. BAND-DOMINATED OPERATORS ON LP (R,Y ) 93

0 Hence every operator A p is of the form A = bI + B with B p and if bI is invertible then the assumptions of Theorem∈ B 4.13 and its generalizations which∈ were B mentioned above hold. In particular, the Fredholm property and the Fredholm index of A are determined by its limit ˜ ˜ 0 operators. For this notice that A can be written in the form A = bI(I + B) with B p, and the snapshots of bI are invertible. ∈ B Of course, also Theorems 4.11 and 4.12 apply, telling us everything about stability, Fredholmness and further asymptotic properties. The paper [4] already deals with such convolution operators and contains results on convergence of norms and pseudospectra. Several applications for boundary integral equations can be found in [16].

4.2.5 A remark on the connection to the discrete case In the above discussion we avoided to exploit the analogies with the lp-case, just to demonstrate seamlessly how the theory works in practice. However, we do not want to suppress this fact, but we shortly demonstrate the idea of discretization which transforms both cases into each other. It is well known and has been frequently applied in the literature ([61], [63], [45], [44], [15]). Also in [81] this path was taken to derive the above results on the Fredholmness and stability. The convergence of the norms, condition numbers and pseudospectra for sequences in ALp appears here for the ﬁrst time. F As a start, we mention that the lp-theory which has been developed step by step in the Sections 2.5 and 3.2.5 can be also built upon the projections

L˜ :(x ) (..., 0, x− , . . . , x − , 0,...) n i 7→ n n 1 without any change or loss in the outcome. p Now, let χ0 denote the characteristic function of the interval I0 := [0, 1) and set X := L (I0,Y ). The mapping G which sends the function f Lp to the sequence ∈

Gf = ((Gf)k)k∈Z, where (Gf)k := χ0U−kf is an isometric isomorphism from Lp onto lp(Z,X). Thus, the mapping

p p −1 Γ: (L ) (l (Z,X)),A GAG L → L 7→ is an isometric algebra isomorphism. Moreover, we have Γ(Lm) = Lˆm and the sets of compact ( -compact) operators translate under the discretization operator Γ. Hence, A (Lp, ) is FredholmP ( -Fredholm, properly -Fredholm, or properly -deﬁcient) if and only∈ if LΓ(A) Pis so. The resultsP of Section 3.1 from [63P] together with TheoremP 1.55 tell us that

Proposition 4.15. Let A (Lp, ) and h = (h ) . The limit operator A exists if and ∈ L P n ∈ H+ h only if the limit operator (Γ(A))h of Γ(A) w.r.t. h exists. Then (Γ(A))h = Γ(Ah). In particular, A is rich if and only if Γ(A) is rich.

p p Moreover, Γ( L ) = l and there is a natural identiﬁcation of sequences A = An ALp A A { } ∈ F with sequences in A p via Γ. F l 94 PART 4. APPLICATIONS

4.3 Multi-dimensional convolution type operators 4.3.1 The operators and their finite sections Functions, spaces and operators Here we consider the spaces Lp(K), p [1, ] of the ∈ ∞ respective (p-integrable or essentially bounded) functions defined on a cone K R2. By a cone ⊂ K (with vertex at the origin) we mean an angular sector in R2, i.e. a set K which includes λy : λ 0 for each y K and such that x K : x = 1 forms a connected closed infinite { ≥ } ∈ { ∈ k k } subset of the unit circle. Clearly, R2 itself is a cone. The papers [49], [50] of Mascarenhas and Silbermann deal with the finite section method for convolution type operators which are introduced as follows: Let u L1(R2) and λ C. We ∈ ∈ denote by C(a) the convolution operator on X := Lp(R2) defined by

C(a): X X, g λg( ) + u( s)g(s)ds, → 7→ · 2 · − ZR where a is given by a(x) = λ + (F (u))(x), with F being the Fourier transform on R2. The function a, usually called the symbol of the operator C(a), is continuous in R2 and tends to λ at inﬁnity. It is well known that C(a) is a linear bounded operator on X, its invertibility is equivalent to the invertibility of the symbol a, and C(a1)C(a2) = C(a1a2) holds for all symbols 3 a1, a2. The set of all such symbols

2 1 2 W (R ) := a = λ + F (u): u L (R ), λ C { ∈ ∈ } ∞ 2 is a subalgebra of L (R ), closed in the norm a W := λ + u L1 , and is called the Wiener 2 k k | | x k k algebra. Let U1(0) be the open unit disc in R and ξ : x be the homeomorphism which 7→ 1−|x| 2 2 2 maps U1(0) onto R . We denote by C(R ) the set of all continuous functions f on R for which f ξ admits a continuous extension f]ξ onto the closed unit disc B1(0). Now,◦ the convolution type operators on◦ a cone K under consideration in [49] (the case p = 2) and [50] (the cases 1 < p < ) have been of the form ∞ χ Aχ I + (1 χ )I K K − K with A belonging to the algebra

2 := alg C(a), fI : a W (R ) and f C(R2) (X). (4.8) A { ∈ ∈ } ⊂ L The present approach permits to recover the results of these papers on the ﬁnite sections and to extend them to the cases p [1, ] as well as to some larger classes of operators. Notice that the pioneering stability results∈ are∞ due to Kozak [39] and that the Fredholmness and invertibility of convolution operators have already been studied in several further publications, e.g. [85], [23], [20], [47], [7]. See also [44], Section 4.2, [72], Section 5.8 and the notes in [13], Chapter 3.

Finite section domains Let Ω R2 be a closed bounded set containing 0 as interior point ⊂ and further assume that for each t δΩ there is a cone Kt with vertex in t, neighborhoods Ut 1 ∈ 2 2 0 0 −1 2 and Vt of t and a C -diﬀeomorphism ρt : R R , such that ρ , (ρ ) are bounded on R and → t t ρ (t) = t, ρ0 (t) = I, ρ (U Ω) = V K . t t t t ∩ t ∩ t 2 0 0 Here we say that Kx is a cone with vertex x R , if Kx = x+Kx, where Kx is a cone with vertex at 0, respectively. Figuratively speaking, the∈ domain Ω looks (locally) like a cone in each of its

3See [44], Section 4.2, for example. 4.3. MULTI-DIMENSIONAL CONVOLUTION TYPE OPERATORS 95 boundary points. This property will make a major contribution to the construction of snapshots which shall capture the asymptotics at the boundary points. But firstly, let us precisely define the finite section method: For n N we let nΩ denote the inflated copy nx : x Ω of Ω and ∈ { ∈ } introduce the finite section projection Tn := χnΩI. Then the finite section sequence of a bounded linear operator A (X) is given by T AT + (I T ) . ∈ L { n n − n } A sequence algebraic framework From the general theory and the above examples we already got an impression, that we will require several snapshots, and it is standing to reason that a “central” snapshot at the origin and further ones at every boundary point of Ω are the natural candidates.

We introduce the operators Pn = χBn(0)I of multiplication by the characteristic function χBn(0) of the closed disc with radius n centered at 0. Obviously, = (P ) gives a uniform approxi- P n mate identity which provides X := Lp(R2) with the -dichotomy (the latter follows from Theo- 2 P rem 1.27). For α R we introduce the operator Vα of shift by α on X ∈ V : X X, (V f)(t) := f(t α). α → α − t Now, for given Ω, t δΩ and the respective diffeomorphism ρt, we define operators Rn : X X and their inverses by∈ the rules → y y (Rt g)(y) := g nρ , ((Rt )−1g)(y) = g nρ−1 n t n n t n 0 and we complete this family with Rn := I. t We set T := δΩ 0 and, for every n N and every t T , let En := X, E := X, and t t −1 ∪ { } t ∈ t ∈ En( ) := (Rn) Vnt V−ntRn. Further, set Ln := I for every n and t, and introduce the algebra T ·of all bounded sequences· A of operators A X for which the snapshots F { n} n ∈ t −t W (A) := -lim En (An) (X, ), t T Pn→∞ ∈ L P ∈ exist. Also the ideals and T are defined as usual (see Section 2.1 et seq.). Notice that the t G J mappings En are algebra isomorphisms and all required assumptions, such as the uniformity and separation conditions (I) and (II), are fulfilled.4 So, until now, we constructed a setting which allows to study finite section sequences TnATn + (I Tn) (which arise from a given finite section domain Ω) by the Fredholm theory{ of Part2. As− a next} step we seek for a localizing setting which should help to obviate the T -Fredholm property in the main results. J A localizing setting Throughout the previous examples we were interested in very large and abstract classes of (band-dominated) operators and the arising finite section sequences. It was necessary to consider the properties of subsequences, and we had to construct the suitable localizing setting for every sequence individually. Here we tread a simpler path, define a localizing setting in the very beginning and commit ourselves to use it for all occurring (sub)sequences. Of course, the price that we have to pay for that is the limitation to sequences which are compatible with this predefined setting. Nevertheless, it will work sufficiently well for all we are interested in. 2 x For each function ϕ C(R ), and for each t > 0, we again define inflated copies by ϕt(x) := ϕ( t ) ∈ 2 T and we consider the set := ϕnI : ϕ C(R ) which belongs to . The central snapshot W 0 ϕ I of ϕ I equalsC {{ϕ(0)I} and,∈ for t δ}Ω, we have W t ϕ FI = ϕ(t)I, since { n } { n } ∈ C ∈ { n } t t −1 s (V−ntRnϕn(Rn) Vnt ϕ(t)I)Pm = sup ϕ ρt t ϕ(t) 0 k − k s∈B (0) − n − → m 4 t To check the uniform boundedness of the Rn apply the estimate (4.13) with ψ ≡ 1. 96 PART 4. APPLICATIONS

as n for every fixed m, and the dual estimate with Pm multiplied from the left holds as well. → ∞ ∗ ∗ Provided with the involution : ϕnI + ϕnI + , the set becomes a commutative C - { G} 7→ { 2 G} C algebra which is isometrically isomorphic to C(R ) and C(B1(0)) since ϕnI = ϕ ∞ = ϕ ξ ∞ k k k k k ◦ k for every n and ϕ, hence ϕ ∞ = lim supn ϕnI = ϕnI + F/G. Their maximal ideal spaces can be identified (see Theoremk k 3.2) and fork our purposesk k{ it} is convenientGk to think of this space as the compactification R2 of R2 with a sphere of points at infinity and with the Gelfand topology. More precisely, a character on C(R2) is either of the form x : f f(x) with x R2, or of the 7→ ∈ form θ∞ : f f]ξ(θ) with θ δB (0). 7→ ◦ ∈ 1 Here we fix M := Ω R2 and to every x Ω T we associate t(x) = 0. Now, we have to choose ⊂ ∈ \ the functions Fx C(R2) which take their values in the interval [0, 1] and are identically 1 in a neighborhood of x∈ Ω, respectively. We do this in such a way that in case x δΩ the support x ∈ x ∈ of F is contained in (Ux Vx) 0 , and in case x int Ω the function F shall vanish on δΩ. ∩ \{ } ∈ T 0 Finally notice that the sequence T = Tn belongs to : Its central snapshot W (T) is the { } L identity, and for t δΩ the snapshot W t(T) is the operator of multiplication by the characteristic ∈ 0 function χ 0 of the cone K . The localizability of is immediate from the definition. Kt t T Consequently, defines a partially localizing setting and the results of Section 3.2.1 apply. Also notice that C E±t(ϕ I) for every ϕ I and every t T . (4.9) { n n } ∈ C { n } ∈ C ∈ 4.3.2 The main result for localizable sequences Theorem 4.16. The set T of all -localizable sequences is a Banach algebra. For every sequence L C T A = An of the form TAT + (I T) with A the following hold { } − ∈ L A is a Fredholm sequence if and only if all of its snapshots are Fredholm. In this case • t lim ind An = ind W (A) (4.10) n→∞ ∈ Xt T and the approximation numbers from the right/left of the An have the α-/β-splitting property with the finite splitting numbers

t t α = dim ker W (A) and β = dim coker W (A). ∈ ∈ Xt T Xt T If one of the snapshots W t(A) is not Fredholm then, for each k N, • ∈ r l lim sk(An) = 0 or lim sk(An) = 0. n→∞ n→∞

The norms A and A−1 converge and 5 • k nk k n k t −1 t −1 lim An = max W (A) , lim An = max (W (A)) . n→∞ k k t∈T k k n→∞ k k t∈T k k

A is stable if and only if all snapshots are invertible. In this case • t t −1 lim cond(An) = max W (A) max (W (A)) . n→∞ t∈T k k · t∈T k k

The (N, )-pseudospectra of the operators An converge w.r.t. the Hausdorﬀ distance to the • union of the (N, )-pseudospectra of the snapshots.

5We again use the convention kB−1k = ∞ if B is not invertible. 4.3. MULTI-DIMENSIONAL CONVOLUTION TYPE OPERATORS 97

Proof. Recall from Section 3.2.1 the deﬁnition of T , the set of all sequences being of the form T TAT + λ(I T) + J with A T , a complex number λ and a sequence J T . Proposition 3.8 provides that− T and T are∈ Banach L algebras and Theorem 3.9 yields the criteria∈ J for the stability and the FredholmL property.T Then we derive the index formula from Theorem 2.14, the splitting property from Theorem 2.13, and the convergence of the approximation numbers in the case of a non-Fredholm snapshot from Theorem 2.9. We check that even deﬁnes a faithful localizing setting. In the case p = the estimate C ∞ (ϕ A ϕ + ψ B ψ )f = max ϕ A ϕ f , ψ B ψ f max A , B f (4.11) k n n n n n n k {k n n n k k n n n k} ≤ {k nk k nk}k k T 2 holds for all f X, all An , Bn , all n N and all ϕ, ψ C(R ) of norm 1 with disjoint support. For 1∈ p < { we} replace{ } ∈ (4.11 F ) by ∈ ∈ ≤ ∞ (ϕ A ϕ + ψ B ψ )f p = ϕ A ϕ f p + ψ B ψ f p k n n n n n n k k n n n k k n n n k ϕ A p ϕ f p + ψ B p ψ f p max A p, B p f p, ≤ k n nk k n k k n nk k n k ≤ {k nk k nk }k k and we conclude from Remark 3.13 that T / is a KMS-algebra with respect to / in all cases T G T C G p [1, ]. We now show that for every A , every g + and every x Ω ∈ ∞ ∈ T ∈ H ∈ x φx(A + ) W (A) Ag + . (4.12) k G k ≤ k k ≤ k Gk We start with the following observation: Let > 0 and t δΩ be given. Then there is a function 2 t ∈ t −1 t ψ C(R ) such that ψ ∞ = 1, ψ(t) = 1, F ψ = ψ and ψn(Rn) RnψnI 1 + for all n. In∈ the case p = thisk followsk from the fact (Rt )−1 =k Rt = 1.kk If 1 p

−1 0 −1 0 t −1 together with (ρt ) (t) = I, the continuity of (ρt ) and an analogous argument for (Rn) give the claim for suﬃciently small support of ψ. We conclude for every x Ω ∈ x x φx(A + ) = φx(A + ) = φx( ψnI A ψnI + ) k G k k xG k k { } { } x G kx ψnI A ψnI + = lim sup ψnEn(W (A))ψnI ≤ k{ } { } Gk n→∞ k k x −1 x x x lim sup ψn(Rn) Vnt(x) W (A) V−nt(x) RnψnI (1 + ) W (A) . ≤ n→∞ k kk kk kk kk k ≤ k k Theorem 1.19 further yields

x −x W (A) lim inf Eg (ψgn Agn ψgn ) k k ≤ n→∞ k n k x x −1 lim sup V−gnt(x) Rgn ψgn I Agn ψgn (Rgn ) Vgnt(x) (1 + ) lim sup Agn . ≤ n→∞ k kk kk kk kk k ≤ n→∞ k k Since was arbitrary, we arrive at (4.12) and thus, in view of Remark 3.13, we have proved that deﬁnes a faithful localizing setting. Now, Proposition 3.11 provides the convergence of C −1 An and An and Corollary 3.12 the convergence of the condition numbers in case A is a stablek k sequence.k k From Proposition 3.16 we conclude that all spaces Et or their duals are complex uniformly convex, hence Corollary 3.19 yields the convergence of the pseudospectra.

We note that the snapshots At = W t(TAT + (I T)) are operators acting on the respective 0 t − t “local” cones K since they are of the form A = χK0 A χK0 I + (1 χK0 )I. t t t − t 98 PART 4. APPLICATIONS

4.3.3 Some members of T L Since T is a Banach algebra, we get a whole closed subalgebra of T and the above results for each ofL its elements for free, if we only have the generators of this algebra.L So, let us start ﬁnding interesting individuals in T . L

Some basic ingredients We have already seen that T is -localizable, and that its snapshots t C T W (T), t δΩ are the projections χK0 I, respectively. Also the identity I belongs to . ∈ t L Furthermore, all sequences A T are -localizable and have -compact snapshots. This particularly includes all sequences∈ JJ withC a -compact operator PJ, and hence all gI arising { } P ∞ 2 { } from operators of multiplication by functions g L (R ) with (I Tn)g ∞ 0 as n . The respective snapshots are W 0 J = J, W 0 gI∈ = gI, and allk other− snapshotsk → vanish. → ∞ { } { } Convolution operators Of course, the most challenging operators here are the convolutions C(a) with a W (R2), and we are going to show that C(a) is in T . Actually, we do not need ∈ { } L to consider all symbols a = λ + F (u) with λ C and u L1(R2) since T is a Banach algebra ∈ ∈ L containing the identity I, but we can assume that λ = 0 and u is continuous with compact support. For the latter notice that every u L1(R2) can be approximated by continuous functions having ∈ 6 compact support and that C(F (u)) u 1 . k k ≤ k kL Proposition 4.17. Let u be continuous with compact support.

2 1. The commutator [ C(F (u)) , ϕnI ] belongs to for every ϕ C(R ). { } { } G ∈ 2. For every > 0 and every t T there is a function ψ C(R2) of norm one, identically 1 ∈ ∈ t in a neighborhood of t and with small support such that ψn(En(C(F (u))) C(F (u))) < for suﬃciently large n. k − k

3. For every compact set M, both χM C(F (u)) and C(F (u))χM I are compact operators.

7 Proof. Let the support of u be contained in the ball B (0). From C(F (u)) u 1 and R k k ≤ k kL x y (ϕnC(F (u)) C(F (u))ϕnI)f(x) = u(x y) ϕ ϕ f(y)dy, − B (x) − n − n Z R using the Hölder inequality, we conclude that ϕ C(F (u)) C(F (u))ϕ I c u 1 , where k n − n k ≤ nk kL x y cn := sup sup ϕ ϕ = sup sup ϕ (r) ϕ (s) . 2 2 x∈R y∈BR(x) n − n r∈R s∈B R (r) | − | n

Since ξ−1(r) ξ−1(s) r s we find | − | ≤ | − | −1 −1 cn = sup sup ϕ ξ ξ (r) ϕ ξ ξ (s) sup sup ϕ ξ (a) ϕ ξ (b) 2 r∈R s∈B R (r) ◦ − ◦ ≤ a∈U1(0) b∈B R (a)∩U1(0) | ◦ − ◦ | n n and the uniform continuity of ϕ ξ yields c 0 as n , the first assertion. ◦ n → → ∞ For the second assertion let t δΩ and notice that, due to the shift invariance of C(F (u)) and the first part, it suffices to show∈ that χ (R−tC(F (u))Rt C(F (u)))ψ I < /2 for every n, k nW n n − n k where W denotes the support of ψ. Set γ := /(4 χ 1 ). k B2R(0)kL 6See [44], Example 1.45, for instance. 7The proofs are modifications of those in [49] and [72]. 4.3. MULTI-DIMENSIONAL CONVOLUTION TYPE OPERATORS 99

For f X and x nW we ﬁnd that χ (R−tC(F (u))Rt C(F (u)))ψ f(x) equals ∈ ∈ nW n n − n

−1 x s s χnW (x) u nρt s ψ nρt f nρt ds u(x s)ψn(s)f(s)ds 2 n − n n − 2 − ZR ZR −1 x −1 y −1 0 y = χnW (x) u nρt nρt det(ρt ) u(x y) ψn(y)f(y)dy. nW n − n n − − Z h i

By Taylor’s Theorem we have ∆ :=n ρ−1(v) ρ−1(w) (v w) t − t − − n ρ−1(v) ρ−1(v) (ρ−1)0(v) (v w) + h(v w)(v w) (v w) ≤ t − t − t − − − − − −1 0 ( ρt ) (v) I + h(v w) n v w , ≤ k − − k | − | with h(v w) 0 as v w 0. Fix δ (0,R). Due to the properties of the diﬀeomorphism k − k → | − | → ∈ −1 0 −1 ρt we can choose W suﬃciently small such that (ρt ) (v) I + h(v w) δ(2R) for all v, w W . Then, for every n and v, w W ,k∆ δ holds− in case−n v k ≤w 2R, and n v w∈ ∆ R if n v w 2R. Moreover,∈ det(ρ≤−1)0( ) is continuous| and− equals| ≤ 1 in t, | | − | − | ≥ | − | ≥ t · that is, by an appropriate choice of W , we can guarantee that 1 det(ρ−1)0( ) γ/( u ) t ∞ on W . Notice that the continuous function u with a compact support− is uniformly· ≤ continuous.k k

Hence the δ (0,R) can be ﬁxed such that ∈ sup sup u(x y + α) u(x y) γ. x,y∈nW |α|≤δ | − − − | ≤ Puzzling all these parts together we obtain for the expressions x y y E (x, y) := χ (x) u nρ−1 nρ−1 det(ρ−1)0 u(x y) ψ (y) n nW t n − t n t n − − n h i that is supported in , and 2 for En (x, y): x, y nW, x y 2R supx,y ∈R En(x, y) 2γ every n. Now, in the cases{ 1 p < ,∈ | − | ≤ } | | ≤ ≤ ∞ p χ (R−tC(F (u))Rt C(F (u)))ψ f p = E (x, y)f(y)dy dx k nW n n − n k n ZnW ZnW p p

E (x, y)f(y) dy dx (2γ)p f(y) dy dx ≤ | n | ≤ | | ZnW ZB2R(x) ! ZnW ZB2R(x) ! p = (2γ)p C(χ ) f p (2γ)p χ p f p = f p, k B2R(0) | |k ≤ k B2R(0)kL1 k k 2 k k and if p = , then we estimate χ (R−tC(F (u))Rt C(F (u)))ψ f by ∞ k nW n n − n k ess sup En(x, y)f(y)dy ess sup 2γ f(y) dy = 2γ C(χB2R(0)) f f . ∈ ≤ ∈ | | k | |k ≤ 2k k x nW ZnW x nW ZB2R(x)

The assertion is trivial for t = 0. For the third part let Br(0) be a ball containing M, ﬁx a continuous function ζ of norm one, equal to 1 on the set Br+R(0) and supported in the set B2r+R(0), and set D := ζC(F (u))ζI. Then χM C(F (u)) = χM D and C(F (u))χM I = DχM I, whereas D is an integral operator with a continuous kernel function which is compactly supported. It is well known 8 that such integral operators are compact.

8For a detailed proof see [63], Lemma 3.2.4, which works for all p ∈ [1, ∞]. 100 PART 4. APPLICATIONS

Together with (4.9) this is everything that we need to directly check the -localizability of A: C All snapshots of A equal C(F (u)), hence are liftable and their liftings also belong to T . We only note that, for all t, τ T , F ∈ E−τ Et (C(u)) C(u) = E−τ Et (C(u) C(u)) + E−τ [C(u) Eτ (C(u))]. n n − n n − n − n The commutation properties and the local equivalence are immediate from the previous proposition and (4.9) as well.

Multiplications by continuous functions For a given function f C(R2) and the sequence ∈ A := fI it obviously holds that W 0(A) = fI. If t is in δΩ then we conclude from the definitions { } that the respective snapshot W t(A) equals (f ξ)(t/ t )I. Now it is easy to check that A T . ◦ | | ∈ L Corollary 4.18. The algebra is embedded in T by taking A as the sequence A . A L ∈ A { } Cones Let a finite section domain Ω be given and let K be a cone with vertex at 0 such that T T A := χK I . We prove A . { }0 ∈ F ∈ L Clearly, W (A) = χK I and, for every x int Ω, the conditions in the definition of localizable ∈ sequences are fulfilled, that is W x(A) are “liftable”, the commutation properties hold and A + , G Ax + are locally equivalent. The cases t δΩ int K with W t(A) = I, and t δΩ K with G ∈ ∩ ∈ \ W t(A) = 0 are obvious as well. So, let t δΩ δK. For the description of the snapshot W t(A) ∈ ∩ we note that there is a uniquely determined half space Ht such that K Br(t) = Ht Br(t) for t ∩ ∩ sufficiently small r > 0, and we claim W (A) = χHt I. t The operators En(χHt I) are operators of multiplication by the characteristic function of a certain set, and they converge -strongly, since A T . Let At denote the limit. Further, all t t P ∈ F P1(Ek(χHt I) Em(χHt I)) are operators of multiplication by a characteristic function. For all sufficiently large− k, m, say k, m N, their norms are less than one, due to the convergence to At, ≥ t t hence they are zero. Consequently, Pn(Ek(χHt I) Em(χHt I)) = 0 for all n and all k, m nN, t t − t ≥ which implies Pn(Ek(χHt I) A )) = 0 all n and all k nN, therefore A must be an operator of − ≥ t multiplication by a characteristic function as well. One now easily checks that A = χHt I, and A proves to be -localizable. The situation isC analogous for cones K with vertex at 0 which comply with the weaker condition T t 0 A := χK Tn . In this case W (A) = χK χHt I for t δΩ δK. { } ∈ F t ∈ ∩ Example 4.19. We want to point out at this juncture that sequences arising from the operators in belong to T for every finite section domain Ω, whereas the operators of multiplication by theA characteristicL function of a cone require a suitable choice of the finite section domain and the rectifying diffeomorphisms ρ , at least for the points in δΩ δK. Here are two examples: x ∩ 1. Of course, it is rather obvious that Ω = B1(0) can be provided with diffeomorphims ρt for every t δΩ which act as the identity on the line through t and the origin, and which make Ω appear∈ locally as a half space in t. Then every cone K with vertex at the origin T defines a -localizable sequence χK I . In particular, for every operator A with A C { } { } ∈T L and its compression AK := χK AχK I + (1 χK )I to the cone K we get AK as well, hence we can apply the finite section− method for such operators acting{ } on ∈ cones:L A := T AK T + (I T). The snapshots at all t “outside” the cone are the identity, for t { } − t t from the interior of K we obtain W ( ) = χ 0 W A χ 0 I +(1 χ 0 )I, that are operators A Kt Kt Kt 0 { } − acting on the respective cones Kt which are half spaces in this particular case Ω = B1(0). Solely the snapshots at the points t δΩ δK are operators acting on rectangular cones, 0 ∈ ∩ namely Kt Ht. To avoid this phenomenon, and to ensure that all snapshots at points t δΩ become∩ operators on half spaces, one could make another construction as follows: ∈ 4.3. MULTI-DIMENSIONAL CONVOLUTION TYPE OPERATORS 101

K K

δΩ δΩ

Figure 4.1: Two possible ways how to treat operators on a cone.

2. For a given cone K we choose Ω such that its boundary δΩ is a smooth curve in every point t δΩ int K, merges into the boundary δK of K, and is a circular arc outside K ∈ ∩ T (see Figure 4.1). The diffeomorphisms can be chosen in such a way that χK Tn . Now, for an operator A with A T the compression A := χ Aχ {I + (1} ∈χ F)I { } ∈ L K K K − K has a -localizable finite section sequence T AK T (I T) whose snapshots are either trivialC or operators on half spaces for every{ t } δ−Ω. We− particularly note again that t ∈ 0 W χK Tn = χK χHt I for t δΩ δK. { } t ∈ ∩ The advantage of the latter example lies in the fact that for convolutions on half spaces the invertibility can effectively be checked.

Proposition 4.20. Let a be in the Wiener algebra W (R2), H,H0 be half spaces and K = H H0 ∩ be a cone. Set A := C(a) as well as AH := χH AχH I+(1 χH )I and AK := χK AχK I+(1 χK )I. Then the following are equivalent: a is invertible; A is invertible;− A is Fredholm; A is -Fredholm;− A is invertible; A is Fredholm; A is -Fredholm; A is Fredholm; A is -Fredholm.P H H H P K K P Proof. It is a well known fact that the Fredholmness of AK and A are both equivalent to the invertibility of the symbol a. Its proof goes back to Simonenko [85] and can also be found in [12], 9.57. (for 1 p < , and for p = by duality and the help of Proposition 1.34). ≤ ∞ ∞ It is also clear that the Fredholmness of AK yields that it is properly -Fredholm, hence it implies its -Fredholmness. So, let A be -Fredholm, choose a non-zeroPα δK and consider P K P ∈ -limn V−nαAK Vnα. Due to the shift invariance of A this limit exists and equals AH . Further Pnotice that this limit is invertible by the discussion after Theorem 1.58. The implications (A invertible A Fredholm A -Fredholm) are obvious, and by H ⇒ H ⇒ H P passing to the -strong limit -lim V− A V with β int H we find in the same way that P P n nβ H nβ ∈ (AH -Fredholm A invertible A Fredholm A -Fredholm) holds. EmployingP this argument⇒ a third⇒ time we finally⇒ get thatP A -Fredholm provides itself as an invertible limit, thus (A -Fredholm A invertible a invertibleP A Fredholm). P ⇒ ⇒ ⇒ K On some simplifications and side effects of Theorem 4.16 Let be the (non-closed) B0 algebra of all finite sums of products of convolution operators C(a) with a W (R2), multipli- ∈ cations fI by f C(R2), compact operators J (X, ) and multiplications gI by functions ∞ 2 ∈ ∈ L P g L (R ) for which (I Tn)g n→∞ 0 holds. Further let denote the closure of 0 in (∈X), which obviously includesk − .k The → following observation hasB also been made in [50]. B L A 102 PART 4. APPLICATIONS

Corollary 4.21. Let A 0, AK its compression to a cone K with vertex at 0, and Ω be chosen in such a way that∈ all B snapshots W t χ T , t δΩ, are either trivial or operators of { K n} ∈ multiplication by the characteristic function of a half space. Then, for A := T AK T + (I T), { } − A is Fredholm iff AK is Fredholm. In this case α(A) = β(A) = dim ker AK and ind AK = 0. • A is stable iff AK is invertible. • t t Proof. Let AK be Fredholm. We show that all snapshots A := W (A), t δΩ, are invertible. ∈ Then, since A T , Theorem 4.16 provides the asserted criteria for the Fredholm property and stability, as well∈ Las formulas on the α- and β-number and the index. The snapshots Bt := W t A with t δΩ are invertible by analogous arguments as in the discussion after Theorem 1.58{ }. Further∈ it is easy to see that, for t δΩ K, we have At = I. t ∈ 2 \ t If t δΩ K, then the B are of the form C(at) with some at W (R ), thus every A (which ∈ ∩ t 0 ∈ is the compression of B to the respective half space Kt ) is invertible as well by the previous proposition. Moreover, Theorem 4.16 tells us that, for large n, An is a Fredholm operator. Note that An consists of compact operators, operators of multiplication by certain functions (which commute with T ) and convolutions C(a), a = F (u), u L1. From Proposition 4.17 we conclude that n ∈ TnC(a) and C(a)Tn are compact, hence An = fnI+Cn with an operator fnI of multiplication and a compact operator Cn. Consequently, ind An = 0, which specifies the index formula (4.10).

Corollary 4.22. Let A and AK be its compression to a cone K with vertex at 0. If AK is Fredholm then its index is∈ zero.B

Proof. By Theorem 1.2 we can choose an operator B in a neighborhood of A such that ∈ B0 its compression BK is Fredholm of the same index as AK . Furthermore, we can choose a finite section domain Ω with the respective diffeomorphisms as in the second part of Example 4.19. Then we are in a position to apply the previous corollary to find ind BK = 0.

Finally, as another example, consider a finite section domain Ω being a polygon and all diffeomor- T T phisms ρt being the identical mappings. Define the algebra by the generators C(a) 2 2 E ⊂ L { } with a W (R ), fI with f C(R ) and f ξ being constant on the unit sphere δB1(0), J ∈ { } ∈ ◦ ∞ 2 { } with compact J (X, ), gI where g L (R ) with (I Tn)g n→∞ 0, χK I with ∈ L P { } ∈ k − k → { } cones K which intersect δΩ solely in its vertices, and by the sequence T. Then, for a sequence A = TAT + (I T) T , the snapshots W t(A) for all t in the relative interior of one edge of δΩ − ∈ E coincide. Moreover, if the snapshot W v(A) at a vertex v of Ω is Fredholm then the snapshots W t(A) for all t in the interior of the flanking edges can be obtained as -strong limits of W v(A), hence are invertible. Further, by Theorem 1.19, they fulfill P

t v t −1 v −1 W (A) W (A) and (W (A)) (W (A)) . k k ≤ k k k k ≤ k k For the latter we again refer to the proof of Theorem 1.58. Thus, in this particular setting, we can replace the index set T in Theorem 4.16 by the ﬁnite set which consists of 0 and the vertices of Ω. This was also discussed in [50] and picks up results of Maximenko [51].

4.3.4 Rich sequences The consideration of rich sequences in the present text permits to include some more classes of multiplication operators which require the passage to subsequences, but then provide convenient snapshots again. Here we consider the set T T of all sequences A with the property RL ⊃ L ∈ F 4.3. MULTI-DIMENSIONAL CONVOLUTION TYPE OPERATORS 103

T that every subsequence of A has a g-localizable subsequence Ag, that is Ag . Moreover, we C ∈ Lg let T denote the set of all sequences TAT + λ(I T) + J with A T , λ C and J T . NoteRT that, in contrast to the set T as introduced− in Section 3.2.4∈, we RL now have∈ one common∈ J localizing setting for all subsequencesTR under consideration. Of course, these definitions are much more restrictive, but their greatest advantage is that T and T can straightforwardly be proved to be Banach algebras. Therefore we again onlyRL need toRT investigate some generators. Nevertheless, T is included in the set T , hence the main results of Theorem 3.22 are applicable to allRT sequences in T . TR Thus, for the Fredholm and stabilityRT criteria, the α- and β-numbers and for the formulas which describe the asymptotics of the norms, condition numbers and pseudospectra of each sequence A T we refer to Theorem 3.22 without stating the results again. Here we only get to know some∈ RT further elements in T . RL Very slowly oscillating functions We say that a bounded and continuous function f is very slowly oscillating, if for each x = 0 and each > 0 there is a δ > 0 such that 6 f(rs) f(rx) < for all s U (x) and all r > 0. | − | ∈ δ The set VSO of all very slowly oscillating functions is a Banach algebra and includes C(R2).A basic example for a function f VSO C(R2) is given by f(x) = sin ln x , x 1. ∈ \ | | | | ≥ Proposition 4.23. For every f VSO the sequence A = fI belongs to T with W 0(A) = fI t ∈ { } RL and the other snapshots W (Ag) at the boundary points t δΩ are multiples of the identity. ∈ Proof. Clearly, if we have a T -structured subsequence Ag of A then it follows immediately from T the definition that Ag belongs to with the snapshots as asserted. Lg Therefore, it suffices to show that A is rich. Let h +, and choose a dense and countable 1 ∈ H t1 subset ti i∈N of T . Then there is a subsequence g h such that W (Ag1 ) exists. To see this, { } ⊂ 1 notice that the sequence (f(hnt1))n∈N has a convergent subsequence (f(gnt1)n∈N with limit ft1 due to the Bolzano-Weierstrass Theorem. Now one easily derives from the definition of VSO t j+1 j 1 that W (Ag ) = ft1 I. By this argument we successively find subsequences g g such that tj+1 n s ⊂ W (Agj+1 ) exists. Define gn := g for every n N and find that W (Ag) exists for every s n ∈ in this dense subset ti i∈N of T . For an arbitrary t T there is a sequence (sm)m ti i∈N { } s∈m ⊂ { } which converges to t. One now easily checks that (W (Ag))m converges to an operator B and −t T that B provides the -strong limit of (E (fI)). Thus, Ag . P gn ∈ Fg We note that the snapshots of VSO-sequences are shift invariant, hence the above Corollaries 4.21 and 4.22 stay valid if we enrich the algebras 0 and by very slowly oscillating functions. Also the algebra T permits an extension in this vein.B B Moreover, weE want to point out that the sequences arising from VSO are treatable by arbitrary finite section domains Ω. Finally, we consider a further class of multiplications which depends on the shape of Ω.

Periodic functions Let the ﬁnite section domain Ω be given. A function f C(R2) is said to be Ω-periodic (or periodic with respect to Ω) with period P if, for every t δΩ∈, the functions ∈ ft : [1, ) C, α f(αt) are periodic with period P , that is ft(α) = ft(α + P ) for every α. ∞ → 7→ 2π A simple example for such a function is shown in Figure 4.2, where Ω = B1(0) and P equals 7 .

Let f be Ω-periodic, h be a strictly increasing sequence of positive integers, and decompose each hn = knP +rn +1 with kn Z and rn [0,P ). Due to the Bolzano-Weierstrass theorem there is a ∈ ∈ 104 PART 4. APPLICATIONS

3 −1 3 2

2 1 1 0 0 −1 −1 −2 −2

−3 −3

Figure 4.2: The function f(z) := sin(7 z ) sin(5 arg(z)), z = 0 is Ω-periodic for Ω = B (0). | | 6 1 convergent subsequence of (rn) with limit r, and we denote by g the respective subsequence of h. t Then, for every t δΩ, the operator W (Ag) exists and equals to the operator of multiplication by the continuous∈ function which takes the value f((1 + r + s)t) on the set δK0 + nP + s for Student Version of MATLAB t every n Z and s [0,P ). ∈ ∈ If, in particular, for one t δΩ, the boundary δKt of the local cone Kt contains the origin then ∈ t f proves to be constant along the ray αt, α [1, ), hence W (Ag) = f(t)I. Now, one easily veriﬁes ∈ ∞

Proposition 4.24. For every Ω-periodic function f the sequence A = fI belongs to T . { } RL Remark 4.25. We finally want to mention, without giving details, that bounded and continuous functions f which are constant along each curve δ(αΩ), α > 0, and which are slowly oscillating also define operators A = fI of multiplication that can be tackled by the tools of the present text. Here and in the literature a continuous function is called slowly oscillating if, for every r > 0, sup f(x) f(y) : y U (x) 0 as x . {| − | ∈ r } → | | → ∞ The idea is to check that the arising sequences A are rich, to construct an adapted “blowing- up-procedure” and to define a localizing setting as∈ Fin Proposition 4.10 such that one finally gets A T . ∈ LR The present ideas and proofs for convolution type operators on Lp(R2) also work for analogous operators on Lp(RN ) and even on Lp(RN , Cm), the product of m copies of the space Lp(RN ). The discrete multidimensional setting lp(ZN ,X) is more involved, since it is not possible to rectify the boundary δΩ by diffeomorphisms as in the continuous case. At least for finite sections with respect to polyhedra Ω which have their vertices in ZN there are comprehensive results [62], [60], Section 6.2 in [63], [69] for the case p = 2, and Section 4.1 in [44]. For deeper results on the Fredholm theory for such finite section sequences and its consequences we refer also to [69], which considers the case p = 2 and already provides results on the asymptotics of norms, condition numbers and pseudospectra. An extension of the Fredholm theory to the cases p [1, ] with finite dimensional X can be found in [80]. By the tools of the present text this now∈ can∞ easily be done for arbitrary Banach spaces X, and can be completed by the statements on the norms, condition numbers and pseudospectra, by the index formula and by certain further classes of rich sequences. Notice that Lindner [43] also considered the stability of finite section sequences with respect to more general starlike domains Ω. 4.4. FURTHER RELATIVES OF BAND-DOMINATED OPERATORS 105

4.4 Further relatives of band-dominated operators

At the end of the4th part we return to our prototypic example, the class of band-dominated operators on lp(Z,X), which has been a guide through the whole text, and we touch upon some related families of operators. At that we have two different aims in mind. On the one hand we want to translate what we obtained in the already studied case to these similar settings. We weaken the notion of band-dominated operators into several directions without loss of the essential results on the applicability of the finite section method. Just to give some keywords, we deal with weakly band-dominated, triangular, quasi-triangular and triangle-dominated operators. On the other hand, for more specific families of operators, such as some comfortable classes of Toeplitz operators or slowly oscillating operators, also the conditions in the theorems and the respective conclusions turn into a simpler and more concrete form which we want to expose.

Since this is just to give some links to well known forerunners and to sound out some possible future work, we keep the presentation as brief as possible.

4.4.1 Band-dominated operators on lp(N,X) p p Recall the approximate projection and the sequence (Ln) in (l (Z,X)). Let P (l (Z,X)) P L ∈ L denote the canonical projection onto the subspace of all sequences (xi) whose entries xi vanish for i 0. This subspace can be (isometrically) identified with the space lp(N,X) of all one-sided infinite≤ and p-summable (for 1 p < ) or bounded (for p = , resp.) sequences. For every ≤ ∞ ∞ bounded linear operator A on lp(Z,X) the compression P AP yields a bounded linear operator on lp(N,X), and vice versa for every B (lp(N,X)) we have PBP + Q (lp(Z,X)), where Q := I P . ∈ L ∈ L − p We introduce the algebra lp( ) of band-dominated operators on l (N,X) as the set of all com- A N pressions of band-dominated operators on lp(Z,X), and note that we would obtain exactly the same if we would build upon shifts and operators of multiplication by bounded sequences as generators for the algebra p . Analogously, we take the compressions PL P as the finite Al (N) n section projections, denote them again by L , and let A p stand for the Banach algebra n l (N) which is generated by all finite section sequences of richF band-dominated operators. Then we can completely translate Theorems 2.31, 2.32, 2.33, 2.35 and 3.23 in the sense of this mentioned 0 embedding. Of course, for every A Alp( ) , we have the basic snapshot W (A) which appears ∈ F N in the form PW 0(A)P + λQ, and only one further “direction” which provides relevant snapshots 1 −1 W (Ag) since the snapshots W (Ag) are always trivial. One usually applies the flip operator J, 1 1 given by (xi) (x1−i), to achieve that these W (Ag) can be regarded as operators JW (Ag)J 7→ acting on lp(N,X) as well. −1 1 −1 Instead of using such flipped versions JEn (An)LnJ = JVnAnLnV−nJ of En (An) for the construction of the snapshots at t = 1, one can also employ the operators

p p Wn : l (N,X) l (N,X), (x1, x2, x3,...) (xn, xn−1, . . . , x1, 0, 0,...) → 7→ −1 and set En (An) := WnAnWn to obtain exactly the same outcome. Actually, the second approach has been the initial one [83] and constitutes a cornerstone in the history of the Banach algebra techniques which make up the largest part of this thesis. Before we turn to the class of Toeplitz operators that has been the engine in the development throughout the last decades we shortly examine a class of band-dominated operators on lp(N, C) whose ﬁnite sections possess surprisingly simple Fredholm and stability criteria. 106 PART 4. APPLICATIONS

4.4.2 Slowly oscillating operators

We call a function a l∞(N, C) slowly oscillating if, for every r > 0, ∈ sup a(x) a(y) : y N such that x y r 0 as x . {| − | ∈ | − | ≤ } → | | → ∞ A band-dominated operator A is said to be a slowly oscillating operator if its coeﬃcients are slowly oscillating, that is A is the norm limit of a sequence of band operators having only slowly oscillating coeﬃcients.

Theorem 4.26. Let A be a slowly oscillating operator and A := LnALn be the sequence of { } its ﬁnite sections. Then A is Fredholm if and only if A is a Fredholm operator. In this case α(A) = β(A) = max dim ker A, dim coker A are splitting numbers. { } In particular, the invertibility of A is necessary and suﬃcient for the stability of A. This result on the stability has been established by Roch, Lindner and Rabinovich in [46](p = 2, banded operators) and [68](1 < p < ) for slowly oscillating operators, whereas it is well known for a much longer time for large classes∞ of Toeplitz operators (see the pioneering work in [24], [83] or the comprehensive monographs [12], [13]). For the remaining cases p = 1, Theorem 4.26 is new. Actually, the additional snapshots of the sequences in this Theorem are∞ Toeplitz, and the proof immediately follows if we invest some facts on such operators. Therefore we defer the proof to the end of the next section.

4.4.3 Toeplitz operators with continuous symbol

A one-sided infinite matrix T (a) := (aj−k)j,k∈N with complex entries ai is called Toeplitz matrix. p Suppose that T (a) induces a bounded linear operator on l (N, C) via (xi) ( ai−kxk). 7→ k∈N Then there is a function a L∞(T) on the unit circle T whose Fourier coefficients form the ∈ P sequence (ai)i∈Z, we further denote this operator again by T (a) and call it the Toeplitz operator with the generating function (or symbol function) a. The set of all such symbols a which define a Toeplitz operator T (a) shall be denoted by M p in what follows, and it is well known that, provided with pointwise operations and the norm p p a M := T (a) L(lp(N,C)), M forms a Banach algebra, the so-called multiplier algebra. k k k k p Recall the shift operators Vk and the finite section projections Ln on l (Z, C), as well as the p p canonical projection P which maps l (Z, C) onto l (N, C) and denote the compressions PVkP p and PLnP again by Vk and Ln. Clearly, the shift Vk on l (N, C) is a Toeplitz operator having the trigonometric polynomial t tk as generating function. So, we see that for every trigonometric polynomial a the corresponding7→ Toeplitz operator T (a) is banded. Let Cp denote the closure of the set of all trigonometric polynomials in M p and refer to its elements as continuous symbols. This definition immediately gives that every Toeplitz operator with continuous symbol is band-dominated. It is well known that M 2 = L∞(T) and C2 = C(T), but for p = 2 the equalities must be replaced by proper inclusions “ ”. Moreover, M 1 = C1 = M ∞ = 6C∞ = W is the so-called Wiener algebra W of all functions⊂ a L∞( ) whose Fourier coefficients (a ) fulfill a < . All T i i∈Z i these definitions and results are∈ taken from [12], Chapter 2 for the cases 1 p <| |. The∞ case p = is tackled by its duality to l1 and the observation T (a)∗ = T (a) in [8≤],P for example.∞ ∞ Theorem 4.27. (cf. [12], 2.47. and Coburns theorem [12], 2.38.) Let a Cp. Then the Toeplitz operator T (a) is Fredholm if and only if a does not have any ∈ zeros on T. In this case, T (a) is one-sided invertible and its index equals wind(a, 0), that is minus the winding number of the curve a around the origin. Moreover, a−1 − Cp and T (a−1) is a regularizer for T (a). ∈ 4.4. FURTHER RELATIVES OF BAND-DOMINATED OPERATORS 107

p Theorem 4.28. Let A = An := LnT (a)Ln with a C . Then { } { } ∈ A is Fredholm if and only if a does not vanish on T. In this case the approximation numbers • of the An have the splitting property with

α(A) = β(A) = ind T (a) = max dim ker T (a), dim coker T (a) = wind(a, 0) . | | { } | | Otherwise, all approximation numbers tend to zero as n . → ∞ The norms A and A−1 converge to the limits (with a˜(z) := a(z−1)) • k nk k n k max T (a) , T (˜a) and max (T (a))−1 , (T (˜a))−1 , respectively. {k k k k} {k k k k}

For every > 0 and every N Z+ the (N, )-pseudospectra spN, An of the finite sections • ∈ converge with respect to the Hausdorff distance to the union spN, T (a) spN, T (ã) as n goes to infinity. ∪

A is stable if and only if a does not vanish on T and its winding number is zero. In this • case the condition numbers cond(An) converge to

max T (a) , T (˜a) max (T (a))−1 , (T (˜a))−1 .9 {k k k k} · {k k k k} This theorem has grown to several expansion stages since the middle of the 20th century, and it holds even for much larger classes of Toeplitz operators, such as for piecewise continuous symbols. A beautiful treatise on this topic with a clear and standalone presentation of the theory around Toeplitz operators and its history can be found in the already mentioned monograph [13] by Böttcher and Silbermann. The particular case of banded Toeplitz matrices is subject of the book [8] by Böttcher and Grudsky. For the most complete resource on this topic and its development we refer to the comprehensive monograph [12] by Böttcher and Silbermann. In the present work, Theorem 4.28 is just a special case of what we know for band-dominated operators on lp(N, C) from Section 4.4.1 and Theorem 4.26, taking Theorem 4.27 into account. We also want to record that the smallest closed subalgebra of (lp(N, C)) containing all Toeplitz operators with continuous symbol is of the form L

T (a) + K : a Cp,K -compact , { ∈ P } and the smallest closed subalgebra of containing all respective ﬁnite section sequences equals F L T (a)L + L KL + W LW + G : a Cp,K,L -compact, G . {{ n n n n n n n} ∈ P { n} ∈ G} To check this take [12], 2.14, 7.7, and Proposition 1.20 into account. We close this section with the pending

Proof of Theorem 4.26. Let A be a slowly oscillating Fredholm operator and A = LnALn be its finite section sequence. As in Section 4.4.1 we see that Corollary 2.30 can be translated{ } to the lp(N, C)-case, hence provides that every snapshot of A is Fredholm. Notice that A is rich, 1 due to Corollary 1.60. Let g + be such that W (Ag) exists, and choose a sequence of banded slowly oscillating operators A∈m Hwhich converge to A in the norm as m goes to infinity. Now, pass 1 1 1 2 1 to a subsequence h of g such that W L 1 A L 1 exists, then to a subsequence h of h such hn hn 1 2 { } n that W L 2 A L 2 exists, and so on. Then define h = (hn) by hn := h . It is obvious from { hn hn } n 9The additional operators T (ã) cannot be omitted in general, see [13], Example 7.14 for instance. 108 PART 4. APPLICATIONS

1 k the deﬁnition that the snapshots W Lhn A Lhn are banded Toeplitz operators for every k and 1 { 1 } 1 converge to W (Ah), which equals W (Ag), as k goes to inﬁnity hence W (Ag) is a Fredholm Toeplitz operator with a continuous generating function. Moreover, from Theorem 2.31 we get 1 that 0 = limn ind Lhn ALhn = ind A + ind W (Ag). Therefore

1 1 dim ker A + dim ker W (Ag) = dim coker A + dim coker W (Ag) where, by Theorem 4.27, at least one of these four non-negative integers is equal to zero. Em- ploying Theorem 2.31 again, we ﬁnd the asserted splitting numbers at least for all subsequences 1 Ag which have a snapshot W (Ag), and Theorem 2.32 provides the Fredholm property of A and the α- and β-number of the whole sequence A. Assume that α(A) is not a splitting number for . Then there is a subsequence of such that lim sr (A ) exists and is larger than zero. A Aj A n α(A) jn 1 Now pass to a subsequence Ag of Aj which has a snapshot W (Ag) and gives a contradiction. The same argument works for β(A). The stability criterion follows from Theorem 2.21.

4.4.4 Cross-dominated operators and the ﬂip

p Here, we consider the closed algebra lp of bounded linear operators on l (Z,X) which is gen- X erated by the shift operators Vα, α Z, by the operators of multiplication aI by sequences ∞ ∈ a l (Z, (X)) and, in contrast to the algebra lp of band-dominated operators, additionally ∈ L A by the flip J which maps the sequence (xi) to (x1−i). We refer to such operators as cross- dominated operators, which is reasonable since the matrix representations of finite combinations of the generators look like a “cross” consisting of two orthogonal bands of finite width. For some similar material on an algebra of convolution type operators and a flip we refer to [71]. ±1 Clearly, the approach of Section 2.5 is not appropriate here, since the homomorphisms En as defined there do not provide a ±1-strongly convergent copy (E±1(A )L±1) of the finite P n n n section sequence An := LnJLn associated to the flip operator, not even for any subsequence. Hence, one gets the{ impression} { that} the present setting is more involved. Actually, this is false. To see this, we use a tricky transformation which shuffles the entries of the sequences under consideration:

x k if k is even p p 2 S : l (Z,X) l (N,X), (xi)i∈Z (yk)k∈N with yk = → 7→ x 1 k if k is odd. ( −2 Obviously, this is an isometric isomorphism, and it can now straightforwardly be checked, by considering the generators, that Ψ: A SAS−1 provides an isometric algebra isomorphism 7→ p which translates lp into the algebra lp( ) of band-dominated operators on l (N,X). Also the X A N ﬁnite section projections Ln are transformed into the respective ﬁnite section projections

˜ p p Ln : l (N,X) l (N,X):(yi)i∈ (y1, y2, . . . , y2n+1, 0,...). → N 7→ Thus, all results on the Fredholm property, on stability, on the formulas for the α- and β-numbers and the index, on the asymptotic behavior of norms, condition numbers and on the convergence of pseudospectra are available for cross-dominated operators as well. To be a bit more precise we note that the isometry Ψ translates the operator of multiplication ∞ by the sequence (. . . , a−2, a−1, a0, a1, a2,...) l (Z, (X)) into the operator of multiplication ∈ ∞L by the bounded sequence (a0, a1, a−1, a2, a−2,...) l (N, (X)), and vice versa. Furthermore, ∈ L using dˆ:= (I, 0,I, 0,I, 0,...), Vˆk := PVkP and Lˆ1 := PL1P , we straightforwardly check that

−1 −1 Ψ(V ) = SV S = dˆVˆ− + (I dˆ)Vˆ + Vˆ Lˆ and Ψ(J) = SJS = dˆVˆ− + Vˆ (I dIˆ ). 1 1 2 − 2 1 1 1 1 − 4.4. FURTHER RELATIVES OF BAND-DOMINATED OPERATORS 109

In matrix notation this is 0 0 I 0 I I 0 0 0 I 0 0 0 0 0 0 I   0 0 I   I 0 0 0 0   I 0 0  Ψ(V1) =   , Ψ(J) =   .  0 0 0 0   0 0 I       I 0 0   I 0       .   .   ..  ..         −1 −1 Conversely, Ψ (Vˆ ) = V− JP + J(I P ) and Ψ (Vˆ− ) = JP + PV− J(I P ), i.e. 1 1 − 1 1 − ......  I 0   0 I  I 0 0 I      0 0   0 I    ,   .  0 I   I 0       0 I   I 0       I   0       .   .   . .   . .          4.4.5 Quasi-triangular operators Definition 4.29. Let X be a Banach space. Let further A (X) and assume that there is a bounded sequence (L ) (X) of projections, such that ∈ L n ⊂ L (I L )AL 0 as n . k − n nk → → ∞ Then A is said to be quasi-triangular with respect to (L ). If even (I L )AL = 0 holds for n − n n every n then A is called triangular w.r.t. (Ln). For a collection of several basic properties and applications of quasi-triangular operators on Hilbert spaces see [2]. By some straightforward adaptions we find that analogues of Proposi- tion 4.2 and Corollary 4.3 hold for an invertible quasi-triangular operator A whose inverse is quasi-triangular, too. So, we again see that a sequence of projections (Ln) which makes an operator A quasi-triangular is quite well suited to define a finite section method. In what follows, we assume that (L ) itself converges in a convenient manner: Let = (P ) be a uniform n P n approximate identity on X, which equips X with the -dichotomy, and suppose that all Ln are -compact projections which tend -strongly to theP identity. (We do not grow tired of noting P P that the classical situation with compact Ln which converge -strongly, and without a , is covered by our proofs as well.) ∗ P Proposition 4.30. The set QT of all operators in (X, ) which are quasi-triangular P,(Ln) L P w.r.t. (Ln) is a Banach subalgebra of (X, ) and contains QDP,(Ln) the set of all quasi-diagonal operators (w.r.t. (L )) as well as theL idealP (X, ). n K P The proof of this proposition is easy. Furthermore the finite section sequence (LnALn) of each operator A QTP,(Ln) converges -strongly to A since A (X, ) and Ln I -strongly. So, we introduce∈ as usual, namelyP as the closed algebra∈ which L isP generated→ by allP sequences FQT LnALn with A QTP,(Ln). Notice that W (A) := -limn AnLn exists for every sequence { } ∈ P A = An QT by Theorem 1.19, and that W (A) is quasi-triangular. { } ∈ F 110 PART 4. APPLICATIONS

Theorem 4.31. Let A = An QT and suppose that W (A) is Fredholm with a regularizer in { } ∈ F QTP,(Ln). Then the sequence A is Fredholm, α(A) = dim ker W (A) and β(A) = dim coker W (A) are splitting numbers and limn ind An = ind W (A). r l If W (A) is not Fredholm then infinitely many approximation numbers sk(An) or sk(An) tend to zero as n . → ∞ The sequence A is stable if W (A) is invertible and its inverse belongs to QTP,(Ln). It is not stable if W (A) is not invertible. Proof. We take up the sequence algebraic framework T as defined for quasi-diagonal operators F in Section 4.1 and note that A T . ∈ F Firstly, we check that A LnW (A)Ln : This obviously holds true if A is the finite − { } ∈ G section sequence LnALn of a single A QTP,(Ln), hence also for a finite product of the form Π L A L with{ A QT} (where∈ we use that (I L )A L 0), and since is a i{ n i n} i ∈ P,(Ln) k − n i nk → G closed ideal the assertion follows for all A QT . Now, let B QTP,(Ln) be a regularizer for ∈ F T ∈ A := W (A). Then B := LnBLn belongs to as well and { } F A L BL = L ABL L A(I L )BL + (A L AL )L BL n n n n n − n − n n n − n n n n = L + L (AB I)L L A(I L )BL + (A L AL )L BL n n − n − n − n n n − n n n n where L A(I L )BL , (A L AL )L BL and AB I is -compact. We can make { n − n n} { n − n n n n} ∈ G − P an analogous observation for LnBLnAn using the quasi-triangularity of A and we get that A is regularly T -Fredholm, hence Corollary 2.24 together with Theorems 2.13, 2.14, 2.9 and 2.21 give the assertion.J

Notice that the additional condition - the regularizer belongs to QTP,(Ln) - is not redundant p in general, as a simple example demonstrates: The shift operator V−1 on l (N, C) is triangular with respect to the standard sequence (Ln), but its regularizer V1 is neither triangular nor quasi- triangular w.r.t. (Ln). Since all regularizers coincide modulo the ideal of compact operators none of them is quasi-triangular. Even for the case of an invertible operator A in a setting with projections Ln having inﬁnite rank there are counterexamples: Set X = Lp[0, 2] and consider the operators C,D on X given by

f(2x) if x [0, 1] f(2x 2) if x [1, 2] (Cf)(x) = ∈ and (Df)(x) = − ∈ . (0 otherwise (0 otherwise

Clearly, both of them are invertible from the left with the left inverses (C(−1)g)(x) = g(x/2) and (D(−1)g)(x) = g(x/2 + 1), respectively. One easily checks that the following operator B on X3 is invertible CD 0 C(−1) 0 0 B = 0 0 C(−1) and B−1 = D(−1) 0 0 .     0 0 D(−1) 0 CD     Now take A (lp(Z,X)) as the block diagonal operator having the blocks B along its diagonal. ∈ L Clearly, A is invertible and triangular w.r.t. (Ln), but its inverse isn’t. Nevertheless, if all Ln are ﬁnite rank operators, A is quasi-triangular w.r.t. (Ln) and invertible, then A−1 is also quasi-triangular. To see this, note that

L L A−1L AL L A−1 (I L )AL 0 as n , k n − n n nk ≤ k nkk kk − n nk → → ∞ that is, for large n, we get the invertibility of LnALn from the left, and there is a uniformly bounded sequence of left inverses in (im L ). Since dim im L < the L AL are even L n n ∞ n n 4.4. FURTHER RELATIVES OF BAND-DOMINATED OPERATORS 111 invertible with uniformly bounded inverses, hence L L AL A−1L = L AL (L L A−1L AL )(L AL )−1L k n − n n nk k n n n − n n n n n nk L AL L L A−1L AL (L AL )−1L 0. ≤ k n nkk n − n n nkk n n nk → Thus, A−1 QT follows from ∈ P,(Ln) (I L )A−1L A−1 A(I L )A−1L = A−1 L AL A−1L k − n nk ≤ k kk − n nk k kk n − n nk A−1 L L AL A−1L + A−1 (I L )AL A−1L 0. ≤ k kk n − n n nk k kk − n nkk nk → 4.4.6 Triangle-dominated and weakly band-dominated operators Encouraged by the latest observations we ask if we can enlarge the class of band-dominated operators on lp(Z,X) in such a way, that also quasi-triangular operators are included, and (at least some of) the results on the stability or almost stability of the respective finite sections still hold. So, take the usual finite section projections (Ln) of Section 2.5 for the following definition. p Definition 4.32. Let p denote the subset of (l , ) whose elements A additionally fulfill Tl L P lim sup (I Ln)ALn−m = 0. (4.14) m→∞ n>m k − k

The operators in p are called triangle-dominated (with respect to (L )). Tl n Clearly, all band-dominated operators (by Theorem 1.55) as well as all operators which are quasi-triangular w.r.t. (Ln) are contained in lp , and lp easily proves to be a Banach algebra. We know that the band-dominated operatorsT requireT at least three “directions” from which we should look at their ﬁnite section sequences to get enough information: Besides the central 0 snapshot W (A) we need to observe the two critical points where the Ln perform the truncation.

Therefore, we let Tlp denote the smallest Banach algebra which contains all ﬁnite section sequences of rich triangle-dominatedF operators, and we use an algebraic framework T as in F Section 2.5.1 in order to study these sequences. Notice that W 0(A) is triangle-dominated for every A Tlp and, proceeding exactly as in the proof of Proposition 2.28, we ﬁnd ∈ F 0 Proposition 4.33. Let A Tlp be such that W (A) is -Fredholm and has a -regularizer in ∈ F P P T lp , and let g +. Then there is a T -structured subsequence Ah of Ag which is -Fredholm. T ∈ H Jh

Now, also the assertions of Theorems 2.31, 2.32 and 2.33 hold true for all sequences A Tlp ∈ F having the additional property that one -regularizer of W 0(A) (if one exists) automatically 10 P belongs to lp . That means, we have criteria for the stability and Fredholm property of such sequences, andT we know that the asymptotic behavior of the approximation numbers and indices of their entries is closely connected with the kernel dimensions and indices of the snapshots. Actually, the need for the additional condition on the regularizers B of W 0(A) stems from the proof of Proposition 2.28, where we use that for both, W 0(A) and B, the relation (4.14) holds. This can be achieved if we assume that W 0(A) fulﬁlls, besides (4.14), also the dual relation for 0 Ln−mW (A)(I Ln) : k − k Deﬁnition 4.34. An operator A (lp) is called weakly band-dominated (w.r.t. (L )), if ∈ L n lim sup (I Ln)ALn−m , Ln−mA(I Ln) : n > m = 0 m→∞ {k − k k − k } holds. The set of all weakly band-dominated operators shall be denoted by p . Wl 10This even covers the above counterexamples for the “quasi-triangular setting”, but it remains as an open question if, or under which conditions the regularizers of a triangle-dominated operator are of that type again. 112 PART 4. APPLICATIONS

Since this is exactly what we know as -centralized elements from Section 3.3.3, lp proves to p P W be a Banach subalgebra of (l , ). For Wlp , the smallest closed algebra which contains all ﬁnite section sequences of richL weaklyP band-dominatedF operators, we again adapt Proposition 2.28 and take account of Proposition 3.31 to prove the following

0 Proposition 4.35. Let A Wlp be such that W (A) is -Fredholm and let g +. Then ∈ F T P ∈ H there is a T -structured subsequence Ah of Ag which is -Fredholm. Jh This permits to replace p by p in the Theorems 2.31, 2.32 and 2.33 without any further Al Wl restrictions. We note that clearly lp lp by Theorem 1.55 and the inclusion is proper in general as the next section reveals. A ⊂ W

∞ 4.4.7 Toeplitz operators with symbol in Cp + Hp After this short course on generalizations of band-dominated operators on lp(Z,X) we again look at Toeplitz operators on lp(N, C), and we mention that the notions of triangle-dominated and weakly band-dominated operators translate to these spaces in a natural way, as in Section 4.4.1. Here, only the spaces with 1 < p < are of interest, since M 1 = C1 = M ∞ = C∞ = W are already completely resolved. Notice that∞ the set of compact and the set of -compact operators coincide for every 1 < p < , respectively. P ∞ ∞ ∞ Let H stand for the set of all functions a L (T) whose Fourier coeﬃcients fulﬁll ai = 0 ∞ p ∞ 11 ∈ for all i > 0, and set Hp := M H . Furthermore, let χ1 denote the function χ1(x) = x. ∩ ∞ p p ∞ From [12], 2.51 we know “It is obvious that Hp is a closed subalgebra of M . Let alg(C , Hp ) p p ∞ p ∞ denote the smallest closed subalgebra of M containing C and Hp . It is clear that alg(C , Hp ) ∞ iβ coincides with alg(χ1, Hp ). The discontinuous function (1 χ1) (β R 0 ) can be shown ∞ − ∈ \{ } p ∞ to belong to Hp for all 1 < p < . ([12], Theorem 6.42 et seq.) This shows that alg(C , Hp ) is strictly larger than Cp.” Moreover,∞ Sarason [77] proved that this algebra even coincides with

Cp + H∞ := f + g : f Cp, g H∞ , (4.15) p { ∈ ∈ p } and we use this notation in what follows. Böttcher and Silbermann observed that for a function p ∞ p a C + Hp the Toeplitz operator T (a) is Fredholm on l (N, C) if and only if a is invertible in p∈ ∞ −1 C + Hp . In this case T (a ) is a regularizer for T (a). (see [12], Theorems 2.53, 2.60). Also recall Coburns theorem ([12], Theorem 2.38) which yields that every Fredholm Toeplitz operator is one-sided invertible. We condense the previous observations to the following generalization of both Theorem 4.28 and the stability criterion in [12], Theorem 7.20.

p ∞ Theorem 4.36. Let A = An := LnALn with A = T (a) + K, a C + H and K compact. { } { } ∈ p Then A is Fredholm if and only if A is Fredholm. In this case the approximation numbers of the An have the splitting numbers α(A) = β(A) = max dim ker A, dim coker A . Otherwise, all { } approximation numbers tend to zero as n . A is stable if and only if A is invertible. → ∞ p Proof. Clearly, A belongs to lp(N), the set of all triangle-dominated operators on l (N), and if A T −1 1 is Fredholm then the regularizer T (a ) is in lp(N), too. The snapshot JW (A)J, as discussed in Section 4.4.1, is the Toeplitz operator withT the symbol a˜ Cp + H∞, a˜(t) := a(t−1). Hence, ∈ p the Fredholmness of T (a) and Proposition 4.33 imply the Fredholmness of W 1(A), and Coburns theorem applied to JW 1(A)J even provides its one-sided invertibility. The rest is as in the proof of Theorem 4.26. 11 ∞ Analogously, H consists of all functions a with vanishing Fourier coeﬃcients ai, i < 0. 4.4. FURTHER RELATIVES OF BAND-DOMINATED OPERATORS 113

This particularly covers the Toeplitz operators with so-called quasi continuous symbols, that are p p ∞ p ∞ functions in QC := (C + Hp ) (C + Hp ). Notice that such Toeplitz operators are weakly band-dominated and, by [12], 2.80,∩ C2 is a proper subset of QC2. For some further reading we also refer to [76], or Appendix 4 in [52], and the literature cited there. Remark 4.37. We also want to mention that there is an analogon to Sarasons observation (4.15) for band-dominated operators. We denote the set of all rich triangular operators in (lp(N, C), ) L P by ∆lp(N) and note that it forms a Banach algebra. Now, it also holds that

p + ∆ p = alg p , ∆ p , (4.16) Al (N) l (N) {Al (N) l (N)} hence this sum is a Banach algebra as well. To prove this we proceed in the same way as in [12] for the Toeplitz case. We recall the mappings SR which were deﬁned by (1.18) and whose p p compressions SR : (l (N), ) (l (N), ) are well deﬁned, linear and bounded by 2 as well. Proposition 1.56 furtherL yieldsP → that L S (A)Pare band operators and S (A) A 0 as R R k R − k → → ∞ for every band-dominated A. Moreover, SR(A) are triangular whenever A is triangular, and if for h = (h ) the limit operator A exists then there is a subsequence g h such that n ∈ H+ h ⊂ (SR(A))g exists as well (Consider the decompositions hn = cnR+rn with cn N and remainders ∈ 0 rn < R, and choose the subsequence g such that the respective remainders converge). If, in ≤ p particular, A (l (N), ) is rich then all SR(A) are rich. ∈ L P p Having these properties of the mappings SR ( (l (N), )) we can apply a lemma of Zalcman ∈ L L P and Rudin (see [12], 2.52) to prove that lp(N) + ∆lp(N) is a closed linear space. If Ai = Bi + Ti, A R i = 1, 2 with Bi lp(N) and Ti ∆lp(N) then the operators Ai := SR(Bi) + Ti converge to ∈ A ∈ R R A in the norm, respectively. Clearly, A A belong to p + ∆ p and converge to A A as i 1 2 Al (N) l (N) 1 2 R , hence A A p + ∆ p as well, that is (4.16) is proved. → ∞ 1 2 ∈ Al (N) l (N) 4.4.8 A larger class of slowly oscillating operators

We ﬁnally enlarge the class of slowly oscillating operators by calling an operator A lp(N) slowly oscillating if it is rich and all its limit operators are shift invariant, that is ∈ T

VkAhV−k = Ah for all k Z and all Ah σop(A). ∈ ∈ Since for every sequence A := LnALn with a slowly oscillating operator A and every h A the 1 { } ∈ H snapshot JW (Ah)J is a Toeplitz operator, we can plug Coburns theorem (which provides one- sided invertibility for every Fredholm Toeplitz operator) into the results for triangle-dominated operators. In analogy to Theorem 4.26 we then obtain

Theorem 4.38. Let A be a slowly oscillating operator and A := LnALn be the sequence of { } its ﬁnite sections. If A is a Fredholm operator with a triangle-dominated regularizer then A is Fredholm. If A is invertible in lp( ) then A is stable. T N Conversely if A is Fredholm then A is Fredholm and α(A) = β(A) = max dim ker A, dim coker A { } are splitting numbers of A. It is not clear at the moment if the condition on the regularizer/the inverse of A can be dropped in general. At least, there is a positive answer if A = B + K where K is -compact and B decomposes into a ﬁnite product of operators which are one of the following P a Fredholm weakly band-dominated operator, • a Fredholm Toeplitz operator with symbol in Cp + H∞, • p a sum I + C with a triangle-dominated C having norm less than 1, • an invertible quasi-triangular operator. • 114 PART 4. APPLICATIONS Index

, ideal in ...... 40 – Symbols – Gm,n F ΓN, , sublevel set ...... 27 ∗ m,n m A , adjoint operator...... 3 γN , γN , γN , γ, level functions ...... 27, 28 ?, Hilbert adjoint operator...... 3 ±, strictly (in/de)creasing sequences...... 55 A H / , quotient algebra ...... 2 A ...... 56 A J H p , band-dominated operators ...... 87 , ideal of compact sequences ...... 50 AL I p , band-dominated operators...... 34 im A, range of A ...... 3 Al α(A), α-number of A ...... 47, 51 ind A, index of A...... 3 l,u , index of the sequence ...... 47 An , adapted finite section ...... 58, 91 ind A A ( ), -compatible elements ...... 76 ind± A, plus-(minus-)index ...... 91 A(P, P), -centralized elements ...... 79 J, flip operator ...... 105 A(P, C ), P -compact elements ...... 76 j(A), injection modulus ...... 21 BA (PS)K, neighborhoodP of S ...... 27 t, T , ideals of sequences ...... 41 J J β(A), β-number of A ...... 47, 51 JV , embedding map...... 21 Bm(A), Bernstein number...... 21 x, local ideals...... 63 γ J bn, inflating spline...... 74, 88 ker A, kernel of A ...... 3 BP ...... 11, 17, 81 (X), (X, Y), compact operators...... 3 0 K K p, p, convolution operators ...... 92 (X, ), -compact operators ...... 9 B B Kl,u P P BUC, bdd and uniformly cts functions . . . . . 35 Ln , adapted finite section projection. . .58, 91

C, complex numbers ...... 2 Lhn , finite section projection ...... 55 , localizing setting ...... 64 $(lp, ), rich operators...... 38 C L P C(a), convolution operator ...... 92, 94 lp = lp(Z,X), l2, l∞, sequence spaces...... 7 cen , center of ...... 63 Lp = Lp(R,Y ), L2, L∞, function spaces . . . . 87 cokerAA, cokernelA of A ...... 3 T , localizable sequences...... 64 L Com( ), commutant of ...... 64 (X), (X, Y), bounded linear operators . . .2 A A L L cond A, condition number of A ...... 69 (X, ), -compatible operators ...... 9 L P P CP ...... 9, 77 C, maximal ideal space ...... 62 p ∞ M C + Hp , algebra of functions ...... 112 Mm(A), Mityagin number ...... 21 p C(R2), continuous functions...... 94 M , multiplier algebra ...... 106 dH (S, T ), Hausdorff distance ...... 32 N, natural numbers, 1, 2,...... 2 p δij, Kronecker delta ...... 17, 23 P , projection “onto l (N)”...... 105 dist(F,G), distance of sets ...... 35 , approximate projection ...... 8, 76 P D ...... 80 -lim An, -strong limit ...... 10, 80 P P P , algebra of sequences ...... 40 p-lim Mn, partial limiting set ...... 70 F p A p , finite section algebra of BdOs ...... 88 Pm, projection on l ...... 7 F L A p , finite section algebra of BdOs ...... 55 ϕt(x), ϕt,r(x), inflated functions ...... 35 F l ( , ), -convergent sequences...... 80 φx(A), local homomorphism...... 63 F(XA,P ), P -convergent operator sequences . 10 q(A), surjection modulus ...... 21 F T , algebraP P of sequences with snapshots . . . 40 QCp, quasi continuous functions ...... 113 Fx F , local filter function ...... 64 QW , quotient map ...... 21 116 INDEX

R, real numbers ...... 2 band-dominated operator...... 34, 87 R±, non-negative (non-positive) reals...... 2 Bernstein numbers ...... 21 ρ , local diffeomorphism...... 94 t –C– rank A ...... 3 t ∗ Rn ...... 95 C -algebra ...... 62 T , rich sequences ...... 53 Calkin algebra ...... 3 R σm(A), singular value ...... 24 center...... 63 s-lim An, strong limit ...... 6 character ...... 62 r l sm(A), sm(A), approximation numbers . . . . . 21 cokernel...... 3 sp A, spectrum ...... 26 commutant ...... 64 sp A, spN, A, pseudospectra ...... 26 complete w.r.t. -strong convergence ...... 12 P spess A, essential spectrum ...... 26 complex uniformly convex space...... 70 σop(A), operator spectrum ...... 37, 87 condition numbers ...... 69 σstab(A), stability spectrum ...... 57 cone...... 94 T = Tn , truncation projections ...... 64 convergence { } T (a), Toeplitz operator ...... 106 -strong ...... 7 T , rich sequences ...... 73 ∗ -strong ...... 10, 12, 80 TRT , truncated localizable sequences ...... 64 strongP ...... 6 T Uα, shift operator ...... 87 uniform ...... 6 u-lim Mn, uniform limiting set...... 70 –D– Vα, shift operator ...... 34 VSO, very slowly oscillating functions . . . . . 103 dual space ...... 3 Wn ...... 105 2 –F– W (R ), Wiener algebra ...... 94 W t(A), snapshot of A ...... 40 Fredholm operator ...... 3 X0, subspace of X ...... 17 Fredholm sequence ...... 51 Z, integers ...... 2 functional ...... 3 Z±, non-negative (non-positive) integers. . . . .2 –G– –A– Gelfand transform ...... 62 algebra Gelfand-Naimark theorem...... 62 Banach ...... 2 C∗-algebra ...... 62 –H– Calkin...... 3 Hausdorff distance ...... 32 center ...... 63 Hilbert space...... 3 KMS...... 63 multiplier ...... 106 –I– sequence algebras...... 40 ideal ...... 2 with approximate projection ...... 76 maximal...... 62 P approximate projection ...... 8, 76 index approximate identity...... 10, 80 operator ...... 3 equivalent ...... 10, 77 plus-(minus-)index ...... 91 uniform ...... 9, 77 sequence ...... 47 approximation numbers ...... 21 injection modulus ...... 21 inverse closedness ...... 2 –B– invertibility ...... 2 Banach algebra ...... 2 generalized ...... 6 Banach space ...... 2 invertible at infinity ...... 9 Banach-Stone theorem ...... 62 involution ...... 62 INDEX 117

-centralized elements ...... 79 –K– P-compactness ...... 9, 76 P-dichotomy ...... 14 kernel ...... 3 P KMS-algebra...... 63 -Fredholmness ...... 9, 79 P-regularizer ...... 9, 76 P –L– projection ...... 4 pseudospectrum...... 26 limit operator ...... 37, 87 -strong convergence...... 10, 80 localizable sequence ...... 64 P localizing setting ...... 64 –Q– faithful ...... 68 quasi continuous functions ...... 113 locally, globally...... 64 quotient algebra...... 2 –M– –R– maximal ideal space...... 62 range ...... 3 Mityagin numbers...... 21 rank...... 3 rich operator ...... 37, 87 –O– –S– Ω-periodic functions ...... 103 operator ...... 2 sequence adjoint ...... 3 α-, β-number ...... 47, 51 band-dominated ...... 34, 87 almost uniformly bounded rank...... 50 compact ...... 3 -localizable...... 64 convolution ...... 92, 94 compactC ...... 50 cross-dominated ...... 108 Fredholm...... 51 Fredholm ...... 3 index...... 47 Hilbert adjoint ...... 3 T -Fredholm...... 42 invertible at infinity ...... 9 Jnullity, deficiency ...... 47 locally compact...... 91 regular T -Fredholm ...... 47 normally solvable ...... 3 splittingJ property...... 49 of multiplication...... 34 stable ...... 7, 50 -compact ...... 9 T -rich, rich ...... 53 P-Fredholm ...... 9 T -structured ...... 53 projectionP ...... 4 singular values ...... 24 properly -deficient ...... 14 spectrum ...... 26 properly P-Fredholm ...... 14 essential ...... 26 quasi-diagonalP ...... 33, 84 pseudospectrum ...... 26 quasi-triangular ...... 109 splitting property ...... 49 rich...... 37, 87 stability ...... 7, 50 self-adjoint ...... 25 surjection modulus ...... 21 shift...... 34 –T– slowly oscillating ...... 113 Toeplitz ...... 106 Toeplitz matrix ...... 106 triangle-dominated ...... 111 weakly band-dominated ...... 111 –U– operator spectrum...... 37, 87 uniform limiting set ...... 70 –P– –V– partial limiting set ...... 70 very slowly oscillating functions ...... 103 118 INDEX Bibliography

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on the thesis “On some Banach Algebra Tools in Operator Theory” presented by Dipl.-Math. Markus Seidel

(1) This text is concerned with the investigation of operator sequences A = An which typ- ically arise from approximation methods for bounded linear operators. The{ } heart of the present approach is to capture the asymptotic properties of such sequences in terms of so-called snapshots, a family of operators where each of them displays one basic facet of the asymptotics, but together they give a comprehensive picture of A. The snapshots are usually taken by a slight transformation of the sequence and a limiting process. One may interpret this as looking at A from diﬀerent angles, and one is interested in results as The stability of a sequence is equivalent to the invertibility of all snapshots. • A sequence An is almost stable (we say Fredholm) iﬀ all snapshots are Fredholm. • In this case{ the} dimensions of the kernels and cokernels of the snapshots as well as their indices provide information on the splitting property and the indices of the An.

The norms An converge to the maximum of the norms of the respective snapshots. • Analogous conclusionsk k hold for the norms of the inverses and the condition numbers.

The pseudospectra of the elements An approximate the union of the pseudospectra of • all snapshots. (2) The stability plays a crucial role for the applicability of approximation methods which replace a linear equation Ax = b by (hopefully simpler) equations Anxn = bn since, as a rule, it holds that the convergence of bn to b and the convergence of An to A together with the stability of An imply the existence of the solutions xn and their convergence to the solution x of the{ initial} equation. Different types of convergence can be studied. (3) Besides the singular values which are well known and available for operators on Hilbert spaces, there are further geometric characteristics being their natural generalizations to the Banach space case, such as the so-called approximation numbers and the related Bernstein and Mityagin numbers. These characteristics provide the right language to describe the behavior of almost stable (i.e. Fredholm) sequences and constitute the connecting link between the Fredholm properties of the snapshots of an operator sequence A and the { n} Fredholm properties of its entries An. More precisely, the so-called splitting property determines how many approximation numbers (or singular values in case of Hilbert spaces) of the An tend to zero as n gets large. Moreover, the indices of the An converge and the limit is nothing but the sum of the indices of all snapshots. (4) The above observations particularly apply to band-dominated operators on sequence spaces lp or function spaces Lp with 1 p , to quasi-diagonal operators and to convolution type operators on cones. They recover≤ ≤ lots ∞ of well known results for more concrete classes of operators, such as Toeplitz operators, but translate also to larger and more general classes: quasi-triangular, triangle-dominated, or weakly band-dominated operators. (5) In that business it turns out to be extremely fruitful to replace the classical triple (compactness, Fredholmness, strong convergence) by a similar concept which defines the substitutes -compactness, -Fredholmness and -strong convergence for operators on a Banach space XP with the helpP of so-called uniformP approximate identities . One restricts considerations to the set (X, ) of operators A which are compatibleP with -compactness in the sense that the productL P of A with any -compact operator is -compactP again. In this set -Fredholmness means invertibilityP up to -compact operators,P and a sequence con- vergesP -strongly iff, multiplied with a -compactP operator, this always turns into norm convergence.P Often, the approximate identitiesP are intrinsically given by an approximation or discretization method, e.g. a sequence of nested projections, and one may say that the -notions are induced by the underlying method. P (6) Actually, the set (X, ) forms a beautiful and perfectly self-contained universe: It is a closed and inverseL closedP algebra, all -regularizers (the “almost inverses”) of -Fredholm operators are still included, and theP -strong limits of sequences in (X, P) belong to that set again. The notion of invertibilityP at infinity, an alternative wayL howP to generalize the classical Fredholm property, proves to coincide with -Fredholmness. In fact, many concrete settings, such as lp or Lp spaces, provide the mentionedP approximate identities in a very natural way (e.g. sequences of canonical projections), where the classical Fredholm property can be completely embraced by the -notions, and large classes of operators (Toeplitz and Hankel operators, convolution typeP and even singular integral operators, band-dominated and quasi-diagonal operators) perfectly fit into this framework. (7) This alternative -concept covers and generalizes the classical one in large parts, and it mimics most ofP the important interactions between approximation methods and the three basic notions compactness, Fredholmness and strong convergence, which are usually exploited. Nevertheless, it displays its full power in settings where the classical approach fails, e.g. if the discretizations An are of infinite rank, hence not compact in general, or in cases where the strong convergence is not available anymore (such as l∞). (8) The above mentioned Fredholm theory for sequences with a certain asymptotic structure (namely the existence of snapshots) can be embedded into a more general approach which does not require any structure and provides a transcendent notion of Fredholmness. Sur- prisingly, the approximation numbers still provide a characterizing description for this general Fredholm property via so-called α- and β-numbers, an analogon to the splitting property for the structured case.

(9) Between these two models another one can be established. The so-called rich sequences do not need to be structured in the previous sense, but shall have a rich amount of structured subsequences, and hence they strike a balance and beneﬁt from both models. This dramatically enlarges the class of sequences whose asymptotics can be captured by snapshots (or snapshots of subsequences, in this case) but keeps large parts of the desired results.