University of Alberta

INVARIANT SUBSPACES OF POSITIVE OPERATORS AND MARTINGALES IN BANACH LATTICES

by

Hailegebriel Gessesse

A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy

in

Mathematics

Department of Mathematical and Statistical Sciences Edmonton, Alberta Spring 2009 Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition

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Name of Author: Hailegebriel Gessesse

Title of Thesis: Invariant Subspaces of Positive Operators and Martingales in Banach Lattices

Degree: Doctor of Philosophy

Year This Degree Granted: 2009

Permission is hereby granted to the University of Alberta Library to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly, or scientific research purposes only.

The author reserves all other publication and other rights in association with the copyright in the thesis, and except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise re­ produced in any material form whatever without the author's prior written permission.

Hailegebriel E. Gessesse

Date: University of Alberta

Faculty of Graduate Studies and Research

The undersigned certify that they have read and recommend to the Faculty of Graduate Studies and Research for acceptance, a thesis entitled Invariant Subspaces of Positive Operators and Martingales in Banach Lattices submitted by Hailegebriel Gessesse in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Dr. Vladimir Troitsky (Supervisor)

Dr. Alexander Litvak

Dr. Volker Runde

Dr. Nicole Tomczak-Jaegermann

Dr. Valentina Galvani

Dr. Roman Drnovsek (External Reader)

Date: To my Parents ABSTRACT

This thesis is organized into five Chapters. In Chapter 1, we give an in­ troductory part of this manuscript. In Chapter 2, we present preliminaries of the thesis which include basic definitions and some fundamental concepts and facts in functional analysis which are useful for our discussion in later chapters. Chapter 3 deals with the method of minimal vectors and its applications in constructing invariant subspaces of operators on a Banach lattice. For a positive operator T on an ordered Banach space X, one defines (T] = {S ^ 0 : ST < TS} and [T) = {S ^ 0 : ST ^ TS}. We produce invariant subspaces for (T] under some assumptions on T. In Chapter 4, we extend the method of minimal vectors from Banach lattices to ordered Banach spaces satisfying certain conditions and use the extension to construct invariant subspaces of operators defined on ordered Banach spaces with generating cones.

Chapter 5 deals with martingales in Banach lattices. A sequence of contrac­ tive positive projections (En) on a Banach lattice X is said to be a filtration if

EnEm = EnAm. A sequence (xn) in X is a martingale if Enxm = xn whenever n < m. We denote by M(X, (£«)) the Banach space of all norm uniformly bounded martingales. We characterize a Banach lattice for which M(X, (En)) is a Banach lattice. We produce a counterexample that M(X, (En)) is not necessarily a Banach lattice. We also prove that if (En) is dense filtration on a KB-space X, then X is embedded in M(X, (En)) as a projection band. The results in Chapters 3 and 4 have been published (see [29] and [30]). ACKNOWLEDGEMENT

I would like to express my heart felt gratitude to my supervisor, Dr. Vladimir G. Troitsky, for introducing me the problems on which I worked to prepare this thesis and for his numerous suggestions and improvements. I am also deeply grateful for his steadfast encouragement, constructive feedback, and mathematical insights. Our many hours of discussions have given me an invaluable experience that I carry on in my future career. I also thank him for his financial support over these many years. I would also like to thank my professors at the University of Alberta, Dr. Alexander Litvak, Dr. Volker Runde, and Dr. Nicole Tomczak- Jaegermann for all their help during my stay at the Department of Math­ ematical & Statistical Sciences, University of Alberta. Many thanks to my friends and colleagues who were behind me in writing this thesis. I thank my parents for all their encouragement and prayers. Finally, I would like to acknowledge the University of Alberta and all the fund sources for their financial and other supports. Table of Contents

1 Introduction 1

2 Preliminaries 8 2.1 Banach spaces 8 2.2 Sobolev spaces 10 2.3 Banach lattices 11 2.4 Operators 15 2.5 C*-algebras 19 2.6 Invariant Subspace Problem 21 2.7 Lebesgue-Bochner spaces 25

3 Invariant subspaces of operators on a Banach lattice 27 3.1 Minimal vectors in Banach lattices 28 3.2 Application of the method of minimal vectors 35 3.3 Invariant subspaces of compact-friendly operators 39

4 Invariant subspaces of operators on ordered Banach spaces 44 4.1 Ordered Banach spaces 44 4.2 Positive quasinilpotent operators on Krein spaces 51 4.3 Applications to C(K) and Ck(fl) spaces and to C*-algebras . 53 4.4 Applications to Sobolev spaces Wk'p(Q) 54 4.5 A special case: WliP(fi) 56 4.6 Minimal vectors in spaces with a generating cone 59 4.7 Applications of minimal vector technique 64 4.7.1 Non-unital uniform algebras 64 4.7.2 Sobolev spaces 64

5 Banach lattice martingale spaces 67 5.1 Martingales in Banach lattices 67 5.2 Examples of martingales in Banach lattices 69 5.3 When is M(X, (£„)) a Banach lattice? 70 5.4 Counterexamples for non-Banach lattice martingale spaces . . 75

5.5 How does X sit in M(X,(En))? 78

Bibliography 82 Chapter 1

Introduction

The Invariant Subspace Problem is one of the most famous and still unsettled problems in functional analysis and modern mathematics. It is stated as a question: When does a bounded operator on a Banach space have a non-trivial closed invariant subspace? The problem was motivated by the Jordan decom­ position of matrices and by the desire to understand the structure and the geometry of an arbitrary operator. P. Enflo [26] found an example of a bounded operator on a separable Ba­ nach space without non-trivial closed invariant subspaces. C. J. Read [44] presented an example of a bounded operator on l\ without non-trivial closed invariant subspaces. Thus, these examples established that the invariant sub- space problem in its general form has a negative answer. However, for various important classes of Banach spaces and operators the Invariant Subspace Prob­ lem remains open. For instance, it is not known if every bounded operator on a separable Hilbert space (or a reflexive Banach space) has a non-trivial closed invariant subspace. A huge stride towards solving the Invariant Subspace Problem had been made by N. Aronszajn and K. T. Smith [13] in 1954. They proved that every compact operator on a real or complex Banach space has a non-trivial closed invariant subspace. This was a considerable advancement in the sense that it solves the problem for the space of all compact operators which is a very large

1 class of operators. V. Lomonosov [37] later improved this result by replacing an invariant sub- space by a hyperinvariant subspace (a hyperinvariant subspace of an operator T is a subspace which is invariant under every operator commuting with T). He showed that if X is an infinite dimensional Banach space and T is a non­ zero compact operator, then T has a non-trivial closed hyperinvariant subspace. In fact, V. Lomonosov [37] proved more than this. When X is a complex Ba­ nach space and T is not a scalar multiple of the identity and commutes with a non-zero compact operator, then T has a non-trivial closed hyperinvariant subspace. Numerous articles have been written by several authors over the last fifty years regarding the Invariant Subspace Problem. However, no one could come up with a result that fully distinguishes those operators with non-trivial closed invariant subspaces from those without a non-trivial closed invariant subspace. One of the questions still unanswered regarding the Invariant Subspace Prob­ lem is the question of invariant subspaces of positive operators on Banach lattices, which can be posed as follows: does every positive operator on a Ba­ nach lattice have a non-trivial closed invariant subspace? Since every positive operator keeps the positive cone invariant, many believe that this question might have an affirmative answer. However, only few results have been found by imposing additional assumptions on a positive operator so far. One of the main research focus of the author of this thesis has been constructing in­ variant subspaces of positive operators defined on ordered Banach spaces or Banach lattices. So, to put the work of the author of this thesis in perspective, we discuss various results about the Invariant Subspace Problem for positive operators next. The Invariant Subspace Problem, when specialized to positive operators on ordered Banach spaces, possesses some special and desirable features. Some positive operators have non-trivial closed invariant subspaces which are also order ideals. For instance, the following fact was proved by B. de Pagter [18]: every non-zero compact positive operator T which is quasinilpotent has a non-

2 trivial closed invariant ideal which is also invariant under each positive oper­ ator in the commutant {T}'. Later, Y. A. Abramovich, C. D. Aliprantis, and 0. Burkinshaw [5] proved the following result: ifT:X^X is a positive op­ erator on a Banach lattice and there exists a positive operator S on X with the property ST ^ TS, S is quasinilpotent, and S dominates a non-zero compact operator, then T has a non-trivial closed invariant ideal. At several points of the history of the Invariant Subspace Problem, various techniques were invented to tackle the problem. For instance, V. Lomonosov used a technique that involves fixed point theorem while M. Hilden [41] came up with a technique that exploits the properties of a quasinilpotent operator. At the end of the last century, S. Ansari and P. Enflo [12] invented the so-called method of minimal vectors which is a brand new technique to construct invari­ ant subspaces for compact operators on a Hilbert space. G. Androulakis [10] used this technique to prove the existence of a hyperinvariant subspace of a quasinilpotent operator on a super-reflexive space under some additional as­ sumptions. I. Chalendar, J. R. Partington and M. Smith [15] applied the technique to reflexive Banach spaces. Later, V. Troitsky [47] extended the technique to arbitrary Banach spaces and used the method to produce a com­ mon closed invariant subspace for the commutant of an operator satisfying a certain condition. It was R. Anisca and V. Troitsky [11] who found out that the method of minimal vectors also helps to construct invariant closed ideals for some classes of positive operators on Banach lattices. Since one of the main research focus of the author of this thesis is finding invariant subspaces of positive operators using the method of minimal vectors, the paper by R. Anisca and V. Troit­ sky [11] served as a stepping-stone for the author to the next forward step of his research development. So, it is worth discussing the work of R. Anisca and V. Troitsky in detail. For the sake of shorter terminology, we use the word localization which means joint compactness of a collection of operators. More precisely, given a Banach space X, a collection T C L(X), and A C X, we say that T localizes

3 A if for every sequence (xn) in A there is a subsequence (xni) and a sequence (Tj) in JF such that T^x^ converges to a non-zero vector. R. Anisca and V. Troitsky [11] proved the following theorem using the method of minimal vectors. Recall that the left-commutant and right-commutant of an operator T : X -* X on a Banach lattice is given by (T] = {S ^ 0 | TS ^ ST} and [T> = {S > 0 | TS < ST} respectively.

Theorem 1.1. Suppose T : X ^ X is a positive quasinilpotent operator, one- to-one, with dense range on a Banach lattice, and xo ^ 0 and ||xo||> r > 0.

(i) If the set of all operators dominated by T localizes B(xo, r) n [0, x0], then {T] has a common non-trivial closed invariant subspace.

(ii) If [0,T] localizes B(xo,r) D [0, xo], then the collection {T} of operators has a common non-trivial closed invariant ideal.

As consequences of this result, they proved the following two important facts. IfT is a positive quasinilpotent weighted composition operator on CQ(Q) then (T] has a common closed invariant ideal. If T is a one-to-one interval preserving quasinilpotent operator on CQ(Q), then (T] has a common non- trivial closed invariant ideal. In this thesis work, we will show that Theorem 1.1 can be slightly extended as follows, and this extension turns out to be crucial in producing common non-trivial closed invariant subspaces and ideals for various classes of positive operators, such as compact-friendly operators and operators having in their super-left commutant an operator dominating a compact operator.

Theorem 1.2. Suppose T : X —> X is a positive quasinilpotent operator on a Banach lattice, and XQ ^ 0 and ||a?o||> r > 0.

(i) If there exists R E (T] such that the set of all operators dominated by R localizes B(xo,r) D [0, xo], then (T] has a common non-trivial closed invariant subspace.

(ii) If there exists R in (T] such that [0,R] localizes B(x0,r) D [0, xo], then (T] has a common non-trivial closed invariant ideal.

4 We will see several important consequences of Theorem 1.2. One of the important consequence is related to a theorem by J. Flores, P. Tradacete, and V. Troitsky [Theorem 3.10] [27]; they proved that if T is a quasinilpotent pos­ itive operator on a Banach lattice with a quasiinterior point such that some operator in the right-commutant [T) dominates a non-zero AM-compact op­ erator, then [T) has an invariant closed ideal. In this thesis work, we provide a similar result for the left-commutant (T]. An important class of positive operators is the so-called compact-friendly operators. By applying Drnovsek's theorem (see [24]), Y. A. Abramovich, C. D. Aliprantis, and O. Burkinshaw [5] proved that if a non-zero compact- friendly operator T: X —> X on a Banach lattice is quasinilpotent, then T has a non-trivial closed invariant ideal. Moreover, for each sequence Tn in [T), there exists a non-trivial closed ideal that is invariant under T and each

Tn. In addition, they showed that this fact can be improved when X is a Dedekind complete Banach lattice as follows: if a non-zero compact-friendly operator T : X —> X on a Dedekind complete Banach lattice is quasinilpotent, then [T) has a common non-trivial closed invariant ideal. The preceding results have been some of the motivation of the author of this thesis to find out if the results can also hold true for the left-commutant (T]. By applying the method of minimal vectors, we will prove the following fact. If T is a quasinilpotent compact-friendly operator then at least one of the following is true:

(i) for each sequence {Tn} in (T] there exists a non-trivial closed ideal that

is invariant under T and each Tn, or

(ii) {T] has an invariant subspace.

We will also show, the above result can be improved as follows when the space X is Dedekind complete: if a non-zero compact-friendly operator Q on an order complete Banach lattice is quasinilpotent, then (T] has a non-trivial closed invariant ideal.

5 Another focus of the author's research has also been extending the ap­ plication of the method of minimal vectors from Banach lattices to ordered Banach spaces with generating cones, such as C*-algebras and Sobolev spaces. In this direction, we will show that, under certain conditions, Theorem 1.1 can be extended to ordered Banach spaces. We establish several consequences of the extension. In particular, we produce non-trivial closed common invariant ideals for some classes of operators on Sobolev spaces. Another research direction of the author of this thesis has been studying martingales in Banach lattices. This is a functional analytic model of the classical martingale theory. As it is known, martingale theory is a branch of the classical probability theory. Previously, by finding a model for this theory using Banach lattices and operators on Banach lattices, V. Troitsky [48] studied the classical theory of martingales from the perspective of functional analysis.

In this model, he used a sequence (En) of positive contractive projections on a Banach lattice satisfying EnEm — EnAm for each n, m as a filtration in classical probability theory and sequences of vectors (xn) in the Banach lattice satisfying the condition Enxm = xn whenever n ^ m are taken to represent martingales with respect to the sequence of operators (En). This model is useful in the sense that, under common circumstances, the facts of Banach lattices can be transformed to the classical martingale theory. V. Troitsky introduced the order structure of the space of all bounded martingales in [48] by defining the order on a martingale space coordinate- wise. He proved that, in some important cases, the space of all bounded martingales is itself a Banach lattice. For instance, if the underlining space

X is a KB-sp&ce (e.g., it could be Lp(fi) space for 1 ^ p < oo), the space of all bounded martingales is itself a Banach lattice. As a result, he produced the following refinement of an earlier result of Krickeberg: the space of all Li- bounded martingales in L\{P) is a Banach lattice and, moreover, an AL-space. We will find a characterization of a Banach lattice for which the space of all bounded martingales is a Banach lattice. We will also show that, under some conditions, the underlining Banach lattice is embedded in the space of

6 all bounded martingales as a projection band. This means we can identify each element in the Banach lattice by a martingale. We will also construct a counter example of a Banach lattice and a filtration on the Banach lattice for which the bounded martingale space is not a Banach lattice. This was an answer for a question posed by V. Troitsky in [48].

7 Chapter 2

Preliminaries

In this chapter, we visit some fundamental concepts and essential facts in functional analysis that are relevant to this manuscript. For a more thorough dissection of the material, we refer the reader [16, 31, 8, 1, 46, 20]. We start with the notion of Banach spaces.

2.1 Banach spaces

A norm on a real or complex vector space X is a function ||-|| : X —» [0, oo) such that for all x, y G X and AGK (where K is R or C),

(i) ||x|| =0iffx = 0

(ii) ||Ax|| = |A|||x|| R Ik+ l/||^ |N|+ ||i/||

A norraed space is a pair (X, ||-||) where X is a vector space and ||-|| is a norm. This norm induces a metric on the vector space X in such a way that the distance between two elements x, y (E X is given by d(x,y) = \\x — y\\. If X is complete with respect to this metric (i.e., any Cauchy sequence has a limit in the vector space X), then we call the pair (X, ||-||) a Banach space. In short, we say X is a Banach space if there is no ambiguity in the norm.

8 For any subset A of a Banach space, we denote the (norm) closure of A by A. A Banach space is called separable if it is a closure of a countable set. Otherwise, we call it non-separable. A subset A of a Banach space X is called dense if the closure A of A is equal to the whole space X. Otherwise, we say A is non-dense. The following are some of the well known Banach spaces. Lp(fi), 1 ^ p < oo; the space of all equivalence classes of measurable functions on a measure space (0, J7, fi) (see the last section of this chapter) with finite norm ||/|| = (/„ |/p> drf'*. Lcoi/i)', the space of all equivalence classes of measurable functions on a mea­ sure space (Vi,T, p) (see the last section of this chapter) with finite norm ll/H = ess sup|/|.

Co; the space of all sequences (xn) with xn —»• 0 and norm ||(xn)|[ = sup„ \xn\. c; the space of all convergent sequences with norm ||(xn)|| = supn \xn\. lp, 1 < p < oo; the space of all sequences (xn) with finite norm ||(xn)|| = (E~iWp)1/p.

/oo; the space of all sequences (xn) with finite norm ||(xn)|| = supn \xn\. C(K); the space of all continuous scalar valued functions on a compact Haus- dorff space K with ||/|| = supigA- \f{i)\. CQ{K); the space all continuous scalar valued functions which vanish at infin­ ity on a locally compact Hausdorff space K with ||/|| = supteX |/(£)|. Wk>p{VL); Sobolev spaces (see the following section). One of the most important tools to establish several important theorems in Banach space theory is the Baire category theorem. We state the particular case of this theorem which we use later in one of our proofs.

Theorem 2.1 (Baire Category Theorem). Let X be a Banach space. If X is a countable union of closed sets (Ai), i.e., X = (Ji^i Ai, then at least one of the Ai's has non-empty- interior.

9 If X is a real vector space, then the comlexification of X is the complex vector space

Xc = X ®iX = {x + iy: x,y e X}, whose vector space operations are defined by

(zi +iyi) + (x2 + m) = xi + x2 + i(yi + y2)

(a + i(3)(x + iy) = ax — /3y + i((3x + ay).

If X is also a normed space with norm ||-||, then we can extend the norm ||-|| to a norm on Xc via the formula

||z|| = sup ||:rcos0+ ysin0||. 6»6[0,27r] for each z = x + iy G Xc. Observe that

gdl^ll + Ml) ^ H2II ^ M + IMI

for each z = x + iy G Xc. So, if X is a Banach space, then Xc is also a Banach space.

2.2 Sobolev spaces

n Let Q C R and let / G Lloc(Q,). For a given multi-index a, a function v € Lj^Q) is the ath weak derivative of / if

H f (pvdx= (-1) / /£>Q0 dx

for all (f> G Co°(r2). In this case, we write f = Daf. The Sobolev space, denoted by Wk,p(Q,), is defined as the set of all those functions / G Lp{Vt) for which the weak partial derivatives Daf exist and are

10 in Lp(tt) for each multi-index a with |a| ^ k, i.e.,

k p w ' (n) = Lp(n) n {/: DJ e LP(Q), \a\ < fc}.

The norm on Wk,p{Vt) is given by

if 1 ^ p < oo, T.\\Dt = < E l|AJ|l if p = oo,

where ||-||L is the norm in LP(Q). As usual, we will write 1 for xn- For more details on Sobolev spaces, we refer the reader to [51, 31, 43, 28]. One of the celebrated theorem in nonlinear analysis is the Sobolev Em­ bedding Theorem. We will use this theorem later when we deal with order structure of Sobolev spaces. The theorem is stated as follow.

Theorem 2.2 (Sobolev Embedding Theorem). Let 0, be a bounded subset of W1 with C0,1 boundary. For 1 ^ p < oo, the following statements are true.

(i) ifkp < n, then Wk*{tt) ^ Lq{tt), for all 1 ^ q < ^.

(ii) if kp = n, then Wk,p(Q) <—» L9(Cl) for alll ^ q < oo. Moreover, if p = 1 and n = k, then ^'(fi) ^ C(fi).

(Hi) ifkp > n, then Wk*{Q) ^ C(fi). where <-+ represents a continuous embedding, i.e., Y '—> X means each class [a] in Y contains an element b (where [b] G X) and the map [a] —> [b] is continuous.

2.3 Banach lattices

In this section, most of the notations and terminologies are taken from [1]. An order relation ^ on a set AT is a relation on X such that x ^ x for each x € X, x ^ y and y ^ x implies x = y, for all i,j/£l, and for x,y, z £ X,

11 x ^ y and y ^ z implies x ^ z. As usual, the symbol x ^ y is equivalent to

A vector space order on a real vector space X is an order relation ^ on X that is compatible with the linear structure of X in the sense that it satisfies the following two conditions.

(i) If x ^ y, then x + z ^ y + z for each 2 G X.

(ii) If x ^ ?/, then ax ^ ay for each a ^ 0.

An ordered vector space is a real vector space equipped with a vector space order.

If X is an ordered vector space, then the set X+ = {x G X : x ^ 0} referred to as the positive cone of X and the elements of X+ are called positive vectors. A cone is a subset C of a real vector space with the property C + C C C, AC C C for each A > 0, and C D (-C) = {0}. Notice that if C is a cone in a real vector space X, then the binary relation ^ defined as x ^ y whenever x — y € C is a vector space order on X with X+ = C. A positive cone of an ordered vector space is called a generating cone if

X = X+ — X+, i.e., if every vector in X can be written as a difference of two positive vectors. A subset A of a partially ordered vector space is called order bounded if there exist x,y £ X such that x ^ a < y holds for each a G A. A vector y in an ordered vector space X is called an upper bound of a subset A of X ii a ^ y for every a G A. A vector y in an ordered vector space X is called a lower bound of a subset yl of X if j/ ^ a for every a E A. For two elements a, 6 in an ordered vector space X, we write [a, b] = {x E X : a ^ x ^ b}. Sets of this form are called order intervals. A vector lattice is an ordered vector space such that every pair of vectors has a least upper bound (supremum) and a greatest lower bound (in- fimum). Following the standard lattice notation, the supremum and infimum of a pair of vectors {x, y} is denoted by x V y and x Ay, respectively. For an

12 element x in a vector lattice, its positive part, and its negative part, and its absolute value (or modulus) are defined by

x+ — x V 0, x~ = (—x) V 0 and |x| = x V (—x), respectively. We have the following three important identities:

x = x+ — x~ \x\ = x+ + x~ and x+ Ax~~ = 0.

The functions (x,y) —> x V y, (x,y) —> x A y, x —> x+, x —> x~, and x —> |x| are collectively referred to as the lattice operations of a vector lattice. Two elements x,y are called disjoint if |x| A \y\ = 0. It is obvious from x = x+ — x~ that every vector lattice has a generating cone. It is also obvious from x+ A x~ = 0 that x+ and x~ are disjoint. There exist several identities and inequalities which are useful for our future discussion. We summarize some of them as follow.

Theorem 2.3. For arbitrary elements x,y,z of a vector lattice and a ^ 0, the following identities hold.

(i) xVy = -((-x) A (-y)) andxAy = -{(-x) V (-y));

(ii) x-\-y = xf\y + xVy;

(Hi) x + (y V z) = (x + y) V (x + z) and x + (y A z) = (x + y) A (x + z);

(iv) a(x V y) = (ax) V (ay) and a(x Ay) = (ax) A (ay);

(v) | |x| - \y\ | ^ \x + y\^ \x\ + |y|;

(vi) \x V z — y V z\ ^ frrr — 2/1 and \x V z — y \/ z\ ^ \x — y\.

A vector lattice is said to be order (or Dedekind) complete whenever every non-empty subset of the space that is bounded from above has a supre- mum. Similarly, a vector lattice is said to be a-order (or a-Dedekind)

13 complete whenever every non-empty countable subset of the space that is bounded from above has a supremum.

We say that a net (xa) in ordered set S is called increasing and write xa j if a ^ P => xa ^ xp. The symbol xa T £ means xa | and x = supxQ.

Similarly, a decreasing net is denoted by xa J. and xa j x means xa J, and x = inf xa. We say that a net (xQ)aer is order convergent to x and write xa A x if there are two nets (aa)aeyi and (bp)peB such that aa T ^, fyg ! :r and for all a G A, (5 G 5 there exists 70 G T such that for all 7 ^ 70, aQ ^ x7 ^ 6/3.

In particular, we say a sequence (xn) is order convergent to an element x and write xn —> x if there exists two sequences (an), (6n) C .5" such that for all neffwe have an ^ xn ^ 6n and an | 1, 6„ | x. A subset i C S is called order closed if it is closed with respect to this convergence. A subset A of a vector lattice X is said to be solid if |x| ^ \y\ and y G A imply x G A A solid vector subspace of X is called an (order) ideal. The idea/ generated by a non-empty subset A of a vector lattice X is the smallest (with respect to inclusion) ideal containing A; it coincides with the intersection of all ideals that contain A and is given by

n

1(A) = {x G X : 3 xi,..., xn £ A and Ai,..., An G [0, 00) with |x| ^ Y~] -^l^l}- 1=1

If ,4 = {x}, then J(x) := I({x}) is called the principal ideal generated by the vector x. Note that

I(x) = {y G X : 3A ^ 0 such that |y| < A|x|}.

A vector e > 0 is called an order unit if 1(e) = X. The disjoint complement of a non-empty subset A of a vector lattice X, denoted by Ad, is defined by the set of all elements in X which are disjoint with every element in A i.e.

Ad = {x G X : |x| A \a\ = 0 for each a G A}.

14 An order closed ideal is called a band. A band B in a vector lattice X is called a projection band if X = B © Bd. The natural projection generated by a projection band is referred to as a band projection. A vector subspace Y of a vector lattice X is called a sublattice if the lattice operations on Y defined by the order of Y inherited from X coincide with the lattice operations on X. A vector lattice X is called a Banach lattice if it is a Banach space and the order is compatible with the norm in the sense that for x,y 6 X, \\x\\ ^ WvW whenever \x\ ^ \y\. In fact, the preceding condition, ||x|| ^ \\y\\ whenever \x\ ^ \y\, is equivalent to the condition, ||x|| = |||x||| for every x and llxll ^ WvW whenever 0 ^ x ^ y. So, we use this conditions interchangeably. A positive vector e in a Banach lattice X is called a quasi-interior point if

A Banach lattice is called order continuous if its norm is order contin­ uous (i.e., xa A x implies \\xa — x|| —»• 0). A Banach lattice X is called a

KB-space if every increasing norm bounded net in X+ is norm convergent, i.e., if 0 < xa 1 and supQ||xa|| < oo, then there exists x e X such that lima||xa — x|| = 0. Any complex Banach space of the form Xc = X © iX, where X is areal Banach lattice is referred to as a complex Banach lattice. The proof of the following theorem can be found in [8, Theorem 3.7, page 32].

Theorem 2.4. Every band in an order continuous Banach lattice is a projec­ tion band.

2.4 Operators

By an operator we always mean a linear bounded operator between normed spaces, i.e., T : X —> Y is an operator if T(ax + by) = aT(x) + bT(y) for all a, b G K, x,y € X and its operator norm defined by ||T|| = sup{||Tx|| : ||x|| < 1} is finite. For a linear map which is^not bounded, we say linear operator. We will usually use the symbols Q, S and T to denote operators. It is well

15 known that an operator between normed spaces is bounded if and only if it is continuous. An operator T on a vector space X is called a projection if T2 = T. An operator T from a Banach space X to a Banach space Y is called a contraction if \\Tx\\ ^ ||a;|| for every xel. An operator T from a normed space X to a normed space Y is called an isometry if \\Tx\\ = \\x\\. An operator is called quasinilpotent if ||T"||1/n —> 0 as n —>• oo. An operator is called locally quasinilpotent at a vector x if ||T™x||1/n —> 0 as n —> oo. It is well known that an operator is quasinilpotent if and only if it is locally quasinilpotent at every vector. We say that an operator is non-scalar if it is not a scalar multiple of the identity operator. We denote the collection of all bounded operators between normed spaces X and Y by L(X,Y). If Y is a Banach space, then L(X, Y) endowed with the operator norm is again a Banach space. We denote L(X,X) by L(X). When Y is the scalar field K, then we denote L(X,K) by X*. We call X* the dual space of X. We call elements of X* linear functionals or simply functionals. We denote the dual of X* by X**. We call X reflexive if the linear map J : X —> X** defined as J(x) = x, where x(f) = f(x) for each / 6 X*, is a surjective map. We make use of the following separation theorem in our proof later. The proof can be found in [9, p. 202].

Theorem 2.5 ([9]). // the interior of a convex subset A of a Banach space X is nonempty and is disjoint from another nonempty convex subset B of X, then A and B can be properly separated by a nonzero continuous linear functional, i.e., there exists a linear functional f and a positive number a such that f(b) ^ a ^ f(a) for each a € A and b € B.

A net (xa) in a Banach space X is said to converge weakly to x in X if f(xa) converges to f(x) for each / e X*. In this case, we write w-lima xa = x, or, xa —» x. We say that a net (fa) in X* weak* converges to / in X* if fa{x) converges to f(x) for every x & X. In this case, we write w*-lima fa = f, or> fa —> f • It is well known that these two convergence notions induce the so-called weak topology and weak* topology in X and X*, respectively. Given

16 an operator T : X —• Y on a Banach space X to a Banach space Y, the operator, denoted by T* : Y* —> X*, from the dual of Y to the dual of X defined by T*f = / oT is called the dttaZ operator or adjoint operator of T. The proofs of the following two theorems can be found in [8].

Theorem 2.6. For a subset D of a Banach space X the following statements are equivalent:

(i) D is relatively weakly compact.

(ii) For each e > 0, there exists a weakly compact subset W of X with

DCW + eB(0,l).

Theorem 2.7. For a Banach lattice X, the following statements are equiva­ lent:

(i) X is order continuous.

(ii) If 0 ^ xn f^ x hold in X, then (xn) is a norm Cauchy sequence.

(Hi) X is a—Dedekind complete, and xn J. 0 in X implies \\xn\\ J. 0.

(iv) X is an ideal of X**.

(v) Each order interval of X is weakly compact.

We also use the following famous theorem of Alaoglu in our later discus­ sions. For the proof, please see [16].

Theorem 2.8 (Alaoglu). For a Banach space X, the unit ball {/ G X* : ll/H < 1} of X* is weak* compact.

An additive (respectively, multiplicative) semigroup of operators is a collection of operators on a Banach space which is closed under addition

17 (respectively, multiplication) of operators. The commutant of an operator T on a Banach space X, denoted by {T}', is defined by

{T}' = {S eL(X) : ST = TS}.

Obviously, the commutant of an operator is an algebra, an additive semigroup as well as a multiplicative semigroup. An element T £ L(X, Y) is called compact if it maps the unit ball {x £ X : ||:r|| ^ 1} of X into a relatively compact set in Y. An operator T from an ordered Banach space X to an ordered Banach space Y is called a positive operator if Tx ^ 0 whenever x ^ 0. In this case, we write T ^ 0. We write x > 0 if x ^ 0 and x 7^ 0. A positive operator is called strictly positive if Tx > 0 whenever x > 0. For two operators T and S, we write T ^ S if 5 — T ^ 0. It is easy to see that this order makes the collection L(X) an ordered Banach space. As usual, the positive cone is denoted by L(X)+. We use the following theorem (which is a special case of Theorem 1.10 in [8]) in the future.

Theorem 2.9. Let T : X —> X be an operator on a Banach lattice such that sup{\Ty\ : \y\ ^ x} exists in X for each x £ X+. Then the modulus of T exists, and \T\(x) = sup{\Ty\ : \y\ ^ x} holds for each x £ X+.

If S and T are two operators on a vector lattice X, we say that T is dominated by S if \Tx\ ^ S\x\ for every x £ X. Notice that if an operator dominates another operator, then it is automatically a positive operator. We say that an operator T on a Banach lattice is AM-compact if it maps order bounded sets to relatively compact sets. A linear operator S : X —> Y from a vector lattice to a vector lattice is called a lattice homomorphism if S(x A y) = Sx A Sy for all x, y £ X, or equivalently, S(x V y) = Sx V Sy for all x,y £ X, or equivalently, S\x\ = \Sx\ for all x £ X. A vector lattice X

18 is lattice homomorphic to a vector lattice Y if there exists a one-to-one surjective lattice homomorphism from X to Y. A positive operator T on a Banach lattice X is said to be compact-friendly if there exist three non-zero operators R, K, and C on X such that R and K are positive, K is compact, T commutes with R, and C is dominated by both R and K. Compact-friendly operators were first studied in [5]. The following lemma is proved in [1, page 137].

Lemma 2.10. For a (real or complex) Banach lattice X with a quasi-interior point u we have the following.

(i) For each non-zero vector y G I(u) there exists an operator M dominated by the identity operator satisfying M(y) > 0.

(ii) For every v satisfying 0 ^ v ^ u there exists an operator T : X —> X dominated by the identity operator and such that Tu = v.

We make use of the following important theorem later in our proof. The theorem is due to Aliprantis-Burkinshaw. For the proof, see [8, page 276].

Theorem 2.11. If in the scheme of operators between Banach lattices S\ : E —> F', S2 '• F —> G and S3 : G —> H each operator is dominated by a compact positive operator, then S3S2S1 is a compact operator.

2.5 C*-algebras

Recall that a Banach algebra is a Banach space equipped with an algebra structure such that the algebra multiplication is continuous. An element e in an algebra A is called a unit if ae = ea = a for each a G A. An element a in an algebra A is called invertible if there exists an element b in A such that ab — ba = e where e is a unit in the algebra. An algebra A is called unitary if it contains a unit. An algebra A is called commutative if ab = ba for each a,b G A. For an element a in a Banach algebra A, the spectrum,

19 denoted a (a), is the set a (a) := {A G K | (Ae — a) is not invertible} where e is the unit in the algebra. For example, for a Banach space X, L(X) is a unitary algebra, but it is not necessarily commutative. It is easy to see that the algebra generated by an operator T on a Banach space (i.e. the smallest algebra containing T), denoted by P(T), is given by

P(T) = {p(T) : p(t) is a polynomial}.

We assume / = T°. Note that the closure of P(T) in L(X) is a Banach algebra. Let A be a Banach algebra. A norm-preserving conjugate linear map x —> x* from A to A is called an involution if it satisfies x** = x and (xy)* = y*x* for all rr, y G A. A C*-algebra is defined as a complex Banach algebra A with involution x —» rr*, satisfying

||rrx*|| = INI2 for all iel An example of a C*-algebra is L(H) (the algebra of bounded operators on a complex Hilbert space H), the involution is given by the adjoint operator T* of any T € L(H). Another example is C(K) (the algebra of continuous complex functions on a compact Hausdorff space K), the involution / —> /* is defined as complex conjugation: /*(£) = f(t) (t £ K). An element x in a C*-algebra is called self-adjoint if x = re*. The set of all self-adjoint elements in .4 is denoted by Asa- It is well known that Asa is a closed real vector subspace of A, and A = Asa + iAsa- The proofs of the following two theorems can be found in [46].

Theorem 2.12. Let A be a C*-algebra.

(i) The set C = {x G Asa : a(x) C [0, oo)} is a closed cone.

+ (ii) For each x G Asa, there exist two unique elements x G C and x~ G C satisfying x = x+ — x~ and x+x~ = x~x+ = 0.

(Hi) C = {yy* : y e A}

20 The preceding theorem shows that Asa is an ordered Banach space with a generating cone. As a result, A is an ordered Banach space.

Theorem 2.13 (Gelfand-Naimark). Every commutative, unital C*-algebra A is isometrically isomorphic to the (complex) algebra C(K) for some compact space K that is unique up to homeomorphism.

2.6 Invariant Subspace Problem

The Invariant Subspace Problem is one of the most prominent questions in functional analysis and modern mathematics to date. It is posed as a question: When does a bounded operator T on a Banach space X have a non-trivial closed invariant subspace? If T : X —> X is a bounded operator on a Banach space, we call a subspace V of X invariant under T (or invariant subspace of T) if T(V) C V. Since the zero space {0} and the whole space X are always closed invariant subspaces, we are looking for other non-trivial closed subspaces. So, we call V non-trivial if V ^ {0} and V ^ X. We call V hyperinvariant for T if SV C V for each bounded operator S that commutes with T, i.e., if S G L(X) and ST = TS imply that S(V) QV. If T is a collection of operators, then we say that V is a common invariant subspace of T if it is invariant under each operator in T. When X is non-separable, then the Invariant Subspace Problem has an im­ mediate answer as we can see that for i^O, the closure of span{x, Tx, T2x,...} is a closed invariant subspace for T. If T is a scalar multiple of the identity operator, then every subspace is invariant under T. When X is a complex finite dimensional space, T is a finite matrix and has an eigen value. So, it is straightforward to see that the eigenspace of T is a non-trivial closed hyperinvariant subspace for T. In view of the above discussions, the Invariant Subspace Problem should be formulated as:

21 When does a bounded operator on an infinite dimensional separable Banach space have a non-trivial closed invariant subspace? Every operator T : X —> X on a Banach space naturally induces an oper­ ator Tc : Xc —> Xc on the complexification of of X defined by

Tc(x + iy) = Tx + iTy.

Note that if T : X —> X has a non-trivial closed invariant subspace, then

Tc : Xc —• Xc has a non-trivial closed invariant subspace. It is easy to see that an operator T has no closed invariant subspace if and only if the algebra generated by T has no common closed invariant subspace. Furthermore, T has no hyperinvariant subspaces if and only if its commutant (which is an algebra) has no common invariant subspaces. This leads to the Invariant Subspace Problem for algebras. For algebras and semigroups of operators the existence of a common non-trivial closed invariant subspace is of interest even in the non-separable Banach space case. Along this line, it is worth mentioning the famous Burnside theorem.

Theorem 2.14 (Burnside). For a finite dimensional complex Banach space X, the only subalgebra of L(X) without a common non-trivial invariant subspace is L{X).

P. Enflo [26] found an example of a bounded operator on a separable Ba­ nach space without non-trivial closed invariant subspaces. C. J. Read [44] presented an example of a bounded operator on i\ without non-trivial closed invariant subspaces. Thus, these examples established that the Invariant Sub- space Problem does not necessarily have a positive answer. However, for vari­ ous important classes of Banach spaces and operators the Invariant Subspace Problem remains open. Among the many interesting results presented in years, N. Aronszajn and K. T. Smith [13] proved the following theorem. Theorem 2.15 (Aronszajn-Smith). Every compact operator on a real or com­ plex Banach space has an invariant subspace.

22 V. Lomonosov [37] later improved this result by replacing an invariant subspace by a hyperinvariant subspace.

Theorem 2.16 (Lomonosov). If X is an infinite dimensional Banach space and T is compact, then T has a non-trivial closed hyperinvariant subspace.

In fact, V. Lomonosov [37] proved more than this.

Theorem 2.17 (Lomonosov). When X is a complex Banach space and T is not a multiple of the identity and commutes with a non-zero compact operator, then T has a non-trivial closed hyperinvariant subspace.

The theorems of Lomonosov are great advancements in the question of the Invariant Subspace Problem since they considerably increase the class of operators with non-trivial closed invariant subspaces. From the definition, positive operators keep the positive cone invariant. However, it is not known whether every positive operator on a Banach lattice has a closed invariant subspace. The Invariant Subspace Problem for posi­ tive operators on ordered Banach spaces possesses some special and desirable features. Some positive operators have not only non-trivial closed invariant subspaces but also have non-trivial closed order ideals. With regard to this, B. de Pagter [18] proved the following theorem.

Theorem 2.18 (de Pagter). Every non-zero compact positive operator T on a Banach lattice which is quasinilpotent has a non-trivial closed invariant ideal which is also invariant under each positive operator in the commutant {T}'.

The following result is due to Y. A. Abramovich, C. D. Aliprantis, and O. Burkinshaw [5] which is an extension of de Pagter's theorem:

Theorem 2.19 (Abramovich-Aliprantis-Burkinshaw). IfT:X—>.Xisa positive operator on a Banach lattice and there exists a positive operator S on X with the property ST ^ TS, S is quasinilpotent at some XQ > 0, and S dominates a non-zero compact operator, then T has a non-trivial closed invariant ideal.

23 R. Drnovsek [24] later extended the preceding result to families of opera­ tors. Before stating his results, we introduce some notations and terminologies.

For a subset D of a Banach space, we let ||D|| = supxeD||x||. For a family C of operators in L(X) and for each n G N, we denote

n C = \C\C2 • • • Cn : C\,... ,Cn G C),

n C x = {CXC2 ...Cnx:Cu...,CneC}ioT each iel

We denote

[C) = {Ae L(X)+ :AC-CA^0 for each C G C},

(C] = {Ae L(X)+ : AC - CA ^ 0 for each C G C},

C' = {Ae L{X) :AC = CA for each C G C}.

When C is a singleton, say C = {T}, we denote [{T}) and ({T}] simply by [r> and (T], respectively. We call [T) = {A G L(X)+ : AT - TA ^ 0} the super-right commutant of T and we call (T] = {A E L(X)+ : AT — TA < 0} the super-left commutant of T. C is said to be locally quasinilpotent at a point x G X if lim ||Cnx||1/n = 0 n—>oo where n ||C x|| = sup{||CiC2... Cnx\\ :C1,...,CneC}

We call C finitely quasinilpotent at x G X if every finite subcollection of C is quasinilpotent at x. We are now ready to state Drnovsek's theorems [24] which have useful consequences in constructing invariant subspaces for compact-friendly operators. It also generalizes the preceding theorem.

Theorem 2.20 (Drnovsek). If a collection C of positive operators on a Banach lattice

(i) is finitely quasinilpotent at some positive non-zero vector, and

24 (ii) some operator in C dominates a non-zero AM-compact operator then C and [C) have a common non-trivial closed invariant ideal.

Theorem 2.21 (Drnovsek). Assume that a non-zero collection C of positive operators on a Banach lattice:

(i) is finitely quasinilpotent at some positive non-zero vector and

(ii) some operator in C dominates a non-zero compact operator.

Then C and [C) have a common non-trivial closed invariant ideal.

2.7 Lebesgue-Bochner spaces

Given a set fl and a collection T of subsets of Q, J7 is called a a-algebra if Q, £ T and the collection is closed under the operation of set complement and countable union, i.e., Ac G J7 whenever A G J7, and for a countable collection of sets {Ai}^ in J7, (J°^ A, G J7. A measure a is a non-negative set function on a cr-algebra such that /z(0) = 0 and u(\J°^1 At) = Yl^i A*(A) for a disjoint 7 7 collection of sets {Aj}^x in J . We call the triple (D^J ,^) a measure space; 7 if /JL(0) = 1, we call the triple a probability space and denote it by (f2, J , P). We call fj, a finite measure if [i(Cl) < oo. Recall that a function / : Q, —> R is called measurable if /_1([a, oo)) is an element of J7 for each a G K. It is well known that, for 1 ^ p < oo, the set

{/ : / is measurable and / \f\p d/j, < oo}

p 1 7 is a Banach space with norm ||/|| = (Jn \f\ dfj,) ^, and denoted by LP(Q, J , u), or simply Lp(u). When p = oo, the set

{/ : / is measurable and ess sup|/| < oo}

is a Banach space with norm ||/|| = ess sup|/|, and denoted by L00(/i).

25 Let X be a Banach space and (Q, J7, fx) be a finite measure. We call a function / : Q, —> X simple if there exist x\,X2,--.,xn in X and Ai,A2,...,An 7 x in J such that / = ]T"=1 iXM where x^W = 1 if * e A, and XAt{t) = 0 if t ^ Af. A function / : Vt —> X is called /i- measurable if there exists a sequence of simple functions (/„) with limn||/n — /|| = 0 /^-almost everywhere. A //-measurable function / : O —> X is called Bochner integrable if there exists a sequence of simple functions (/„) such that

lim [\\fn-f\\dfji = 0.

In this case, fA f d/j, is defined for each A G T by

/ / d/u = lim / /n d/x, where the integral of a simple function is defined in the obvious way as

n A I A £"=i ^XA* dfi = ELi ^(^i )-

: If 1 ^ p < oo, the symbol LP(Q,J ,IJ,;X), or in short Lp(/i;X), will stand for all (equivalence classes of) //-Bochner integrable functions / : 0 —• X such that ll/ll = (/ll/ll^)1/p<°o. P Jo.

With norm ||-||p, Lp(Cl,X) is a Banach space. The symbol L^Cl, T, fi, X), or in short L00(//,X), will stand for all (equivalence classes of) essentially bounded /x-Bochner integrable functions / : Q —> X. With a norm defined by

H/lloo = ess sup||/||x,

Loo(0,,X) is a Banach space. For more details of this section, we refer the reader to [20].

26 Chapter 3

Invariant subspaces of operators on a Banach lattice

There are several " standard" techniques to construct an invariant subspace for an operator on a Banach space or an invariant ideal for a positive operator on a Banach lattice. In particular, fixed point theorems and spectral theory are commonly used for these purposes. In this chapter, we discuss the method of minimal vectors on Banach lattices and its use to produce invariant subspaces and invariant ideals of some type of operators. The method of minimal vectors was invented by S. Ansari and P. Enflo [12] in 1998 to produce invariant sub- spaces for some classes of operators on a Hilbert space. In particular, they used the method to reproduce an earlier result due to Lomonosov which asserts that every compact operator on a Hilbert space has a non-trivial closed invariant subspace. Later, G. Androulakis [10] extended the method to produce invari­ ant subspaces of operators on super-reflexive spaces. In 2004, the method was extended to reflexive spaces by I. Chalendar, J. Partington, and M. Smith [15]. In the same year, V. Troitsky [47] extended the method to arbitrary Banach spaces to produce a common non-trivial closed invariant subspace for a class of operators satisfying a certain condition. In 2005, R. Anisca and V. Troitsky [11] came up with a new idea of applying the technique to produce invariant subspaces and invariant order ideals for a collection of operators on a Banach

27 lattice. Throughout this chapter, we focus on the technique developed by R. Anisca and V. Troitsky. In the sequel, we discuss how this technique helps to construct a non-trivial closed invariant subspace or ideal for a collection of operators.

3.1 Minimal vectors in Banach lattices

Following [47] and [11], the method of minimal vectors and its use in producing invariant subspaces of an operator T on a Banach spaces X can generally be outlined as follows. Let T : X —> X be a quasinilpotent operator on a Banach space. Without loss of generality, we assume that T is also a one-to-one operator with dense range since the ultimate goal is to produce an invariant subspace for T. We take a set A C X such that 0 ^ A and int>l ^ 0. We fix e > 0. Then for every n, 0 ^ T~n(A) since intA ^ 0. Then there exists n yn G T~ (A) for each n such that

n ||T/n||<(l + e)dist(0,T- (yl)).

n We say that (yn) is a sequence of (1 + e) — minimal vectors. Notice that (T yn) is a sequence in A. If for a certain sequence (Sn) of operators, (Snyn) has a subsequence that converges to a non-zero vector, then one can produce an invariant subspace for T. The above is a general outline for any arbitrary Banach space. Next, we will see a particular case when our space is a Banach lattice. We will also choose an appropriate set A which is useful for our subsequent application of the method. Through out this chapter, X is a real Banach lattice with positive cone

X+. The symbol B(x,r) stands for the closed ball of radius r centered at x.

Let x0 G X+ with ||:ro|| > 1. Let B(x0,1) + X+ be the algebraic sum of the

28 two sets, i.e.,

B(x0,1) + X+ = {x + h | x G B(x0,1), h ^ 0}.

Lemma 3.1. For z G X and XQ G X+ with \\XQ\\ > 1, the following are equivalent.

(i) z G B(x0,1) + X+;

(M,) z ^ x for some x G B(x0,1);

(Hi) x0 A Z G f?(xo, 1);

^ ||(xo-z)+||^l.

Proof The equivalence (i)-^(ii) is trivial; indeed if z = x + y for some x G

B(xo, 1) and ?/ G X+, then z = x + y ^ x. Conversely, if z ^ x for some x G B(x0, 1), then 2 — x G X+. So, by writing z = x + (z — x), we get the result. To show (iii)<=>(iv), first, we notice that a — aAb = (a — b)+ is true for any a,b G X. Indeed, by Theorem 2.3 (i) and (iii), we have

a - (a A b) = a + ((-a) V (-6)) =

(a - a) V (a - b) = 0 V (a - b) = (a - 6)+.

+ Thus, ||(x0 — 2) || = ||xo — XQ A Z\\. This shows (iii)o-(iv).

(iii)=>(ii) follows from the fact that z ^ x0 A z.

To show (ii)=>(iii), suppose that z ^ x for some x G 5(x0,1). Then

XQ A X ^ XQ A Z ^ XQ, SO that

+ 0 ^ X0 — Xo A Z ^ XQ — XQ A X ^ (Xo — x) ^ |x0 — x|,

hence, by monotonicity of the norm, ||x0 — x0 A z\\ ^ ||x0 — x|| ^ 1. D

Corollary 3.2. For XQ G X+ with ||x0|| > 1, the set B(x0,1) + X+ is closed, convex, and does not contain the origin. 29 Proof. The set B(x0,1) + X+ is clearly convex as it is an algebraic sum of two convex sets. By Lemma 3.1, 0 ^ -B(x0,1)+X+. Otherwise, if 0 G B(XQ, 1)+X+, then by the fact that x0 € X+ and Lemma 3.1(iv),

W = ll(a:o)+|| = ||(xo-0)+Kl

which is a contradiction to the assumption ||xo|| > 1. To show B(x0,1) + X+ + is closed, first we show that the map / : X —> R defined by z H-> ||(X0 — -z) || is continuous. Indeed, suppose xn —> x. Prom Theorem 2.3(vi),

+ + |(x0 - xn) - (x0 - x) | = |(x0 - xn) V 0 - (x0 - x) V 0| ^

|(x0 - rc„) - (x0 - x)| = \xn - x\.

Thus, by triangular inequality and monotonicity of norm,

+ + < |/(xn)-/(x)|= ||(x0-xn) ||HI(£0-*) ||

+ + ||(x0 - xn) - (x0 - x) \\ <, \\xn - x||.

Thus, f(xn) —> f(x) as xn —> x. Now, let xn be a sequence in B(xQ, 1) + X+ such that xn —> x for some x & X. We show x e 5(x0,1) + A+. Indeed, since + xn G B(XQ, 1) + X+, it follows from Lemma 3.1(iv) that ||(x0 — ccn) || < 1 for + each n. Since f(xn) —> f(x), we get ||(xo —£) || ^ 1. So, again by Lemma 3.1, we have x £ B(xo, 1) + .X+. Therefore, B(XQ, 1) + X+ is closed. •

Lemma 3.3. Suppose T : X —• X be a positive operator such that the ideal

/(RangeT) generated by Range T is dense in X. Let xo £ X+ with ||xo|| > 1. 1 ThenT~ (B(x0,1)+X+) is convex, closed, non-empty, and doesn't contain the 1 1 origin. Moreover, i/zGT" (B(i0ll) + X+), then \z\ £ T~ (B(x0, 1) + X+).

_1 Proof. Since T is a bounded linear map, T (B(x0,1) +X+) is clearly convex, closed, and does not contain the origin. Since the ideal generated by Range T

(denoted by /(RangeT)) is dense, it meets B(xo,l). Let x G B(x0,1) D

/(RangeT). Then x ^ Th for some ft, so that Th G B(x0,1) + X+ by

30 1 Lemma 3.1. Hence, Range Tr\(B(x0,1)+X+) is non-empty, so that T (B(xo, 1)+

X+) is non-empty. l Finally, let z G T- {B{x0,l) + X+), then Tz G B(x0,l) + X+. By Lemma 3.1, Tz ^ x for some x G B(xo,l). It follows from z ^ |z| that

Tz < T|2|, so that ar ^ T|,z|. Therefore, by Lemma 3.1, T\z\ G B(x0,1) + X+, 1 so that \z\ eT- (5(x0,l) + X+). •

Lemma 3.4. Lei T : X —> X be a positive operator such that /(Range T) is

_1 dense in X, x0 G X+ with \\x0\\ > 1, and d = inf{||tu|| : w G T (£?(:EO, 1) +

X+)}. Then the following are true:

1 (i) for each e > 0, there exists a positive vector y in T~ (B(x0,1) + X+) such that \\y\\ ^ (l.+ e)d, and

(ii) there exists afunctional f with \\f\\ = 1 that separates the sets T(B(0, d))

and (B(x0,1) + X+), i.e., there exists a functional f with \\f\\ = 1 and c an c a positive real number c such that f\T{B(o,d)) ^ d f\(B(x0,i)+x+) ^ -

1 Proof, (i) Let d = inf{||tu|| : w G T~ (B(x0,l) + X+)}. Fix positive real 1 number e, there exists x G T~ (B(x0,1) + X+) such that ||x|| < (l + e)d. Since x |||x||| = \\x\\, and x G T'^Bixo, 1) + X+) implies \x\ G T~ {B{xQ,\) + X+) by Lemma 3.3, we choose y = \x\. This proves (i). l l (ii) Note that if z G T- (B(x0, l)+X+)n5(0, d), then Xz $ T- (B(x0,1) +

X+) whenever 0 ^ A < 1. It follows that XTz £ B(x0,1) + X+ for every 0 <

A < 1, so that Tz belongs to the boundary d(B(x0,1) +X+) of B(x0,1) + X+. Then

T(B(0,d))n(B(x0,l)+X+) =

T(B{0, d) n T-\B{x0) 1) + X+)) C d(£(x0,1) + X+).

In particular, T(£?(0, d)) and the interior (S(xo, 1) + X+)° are two disjoint convex sets. Since the later of the two has non-empty interior, they can be separated by a continuous linear functional by Theorem 2.5. That is, there

31 exists a functional / with ||/|| = 1 and a positive real number c such that

0 c f\T(B(o,d)) ^ c and /KB(XO,I)+*+) ^ -

By continuity, f\(B(x0,i)+x+) ^ c. •

Definition 3.5. TTie vector y in Lemma 3.4 is called a (1 + e)-minimal vector for T and -B(x0,1) + X+. The functional f is called a minimal functional for T and B{XQ, 1) + X+.

Lemma 3.6. If y is a (1 + e)-minimal vector and f is a minimal functional for T and B{XQ, 1) + X+, then the following are true.

(i) f is positive;

(ii) /(x0) ^ 1;

(Hi) j^f(Ty) < f(x0ATy) < f(Ty);

M Tk\\T*f\\\\y\\^(T*f)(y)^\\T*f\\\\y\\.

(v) f(Ty)^(l+£)\\x0\\;

(vi) IITVH ^ ii±€fM-

Proof, (i) Let 2 G X+ then x0 + Az G -B(x0,1) + X+ for every positive real number A. It follows that /(x0 + Xz) ^ c, so that f(z) ^ (c — /(xo))/A —> 0 as A —> +00. (ii) For every x with ||x|| ^ 1, we have

x0 - x £ B(x0,1) C B(x0,1) + X+.

It follows that f(x0 — x) ^ c, so that /(x0) ^ c + /(x). This implies for any x with ||x|| < 1, we have /(±x) + c ^ /(x0) so that ±/(x) + c ^ /(x0). This implies |/(x)| + c < /(xo). Taking sup over all elements x with ||x|| ^ 1, we get/(x0)^c+H/ll^l.

32 (iii) Since / is positive, it follows from x0 A Ty < Ty that f(x0 A Ty) ^ f(Ty). Notice that y/(l + e) £ B{0,d), so that f(Ty)/(l + e) ^ c. On the other hand, by Lemma 3.1, we have

x0ATy£ B{x0,1) C B(x0,1) + X+, so that

f(x0ATy)>cZ-±-f(Ty). I + £

(iv) We trivially have (T*f)(y) ^ ||r*/||||?/||. Observe that the hyperplane _1 T*f = c separates r (B(i0,1)+^+) and £(0,d). Indeed, if z £ B(0,d), then

(T*f)(z) = f(Tz) < c, and if 2 e T~\B{x^ 1) + X+) then T2 e 5(x0,1) +X+ so that (T*f){z) = /(T2) > c. For every z with ||z|| ^ 1, we have dz £ 5(0,0!), so that {T*f){dz) < c. It follows that ||T*/|| ^ -d- On the other hand, for 1 every 5 > 0, there exists z £ T" {B{xQ, 1) + X+) with p|| ^ d + 5. Then

l whence ||T*/|| > ^r5- It follows that ||T*/|| = % For every z £ T- (B(x0,l) + X+), we have (T*f)(z) ^ c = d\\T*f\\. It follows from ||y|| ^ (1 + e)d that

(T*/)(i/)^itl|T*/||l|y||-

Note that (1 + e)-ly £ B(0,d), hence (1 + e)~lTy £ T(B(0,d)). Since x0 £ B(x0,1) + X+, (ii) and (i) imply

l (1 + e)- f{Ty) ^ c < f(x0) ^ \\f\\\\x0\\ = \\x0\\, which proves (v).

Finally, we get T*f(y) ^ (1 + e)\\x0\\ from (v) and we get ^-||T*/||IMI ^

33 (T*f)(y) from (iv). Combining these two, we get

V^-\\T*f\\\\y\\^T*f(y)^(l + e)\\x0\\.

Therefore, ||T*/|| ^ ^^. •

n Definition 3.7. A sequence (yn) where yn is a (1 +e)- minimal vector for T and B(XQ, 1) + X+ for each natural number n ^ 1 is called a (1 + e)-minimal sequence for T and B(XQ, 1) + X+.

Lemma 3.8. IfT is quasinilpotent and (yn) is a (1 + s)-minimal sequence for T and B(x0,1) + X+, then the following statements are true.

(i) Wl/nW ~* °° as n —> oo.

ni (ii) the sequence (yn) has a subsequence {yni) such that ,, ^ converges to 0.

Proof, (i) Note that, by monotonicity of norm, we have

n n n \\x0AT yn\\^\\T yn\\^\\T \\\\yn\\.

n n Since x0 A T yn e B(x0,1) by Lemma 3.1, we have ||xo|| — 1 ^ ||a^0 A T yn\\, so that ll^oll - 1 < \\Tn \Vn

Since T is quasinilpotent and ||x0|| > 1, we have ||yn|| —• oo as n —> oo. (ii) Suppose the contrary. Then there exists 5 > 0 such that y M > S for all n ^ 2, so that

||yi||^%2||^...^

n n+1 Since T(T yn+1) = T yn+1 £ B(x0,1)+X+, we haveT^n+1 G T~\B(xQ, 1)+

X+). Since 1 d = inf{||iu|| :weT- (5(x0,l)+X+)}, we have

" 1 + £ 1 + £

34 It follows that ||Tra|| ^ 8n/(l + e), which contradicts the quasinilpotence of T. a

3.2 Application of the method of minimal vec­ tors

In this section, we see some applications of the method of minimal vectors to the construction of common invariant subspaces for some classes operators on a Banach lattice. We keep all the notations the same as in the preceding section for the reader's convenience. As usual in this chapter, X represents a Banach lattice. All operators in this section are defined on a Banach lattice. Recall that the super-left commutant and super-right commutant of a positive operator Q are, respectively, defined by

0 | TQ ^ QT} and [Q) = {T > 0 | TQ > QT}.

Lemma 3.9. Let X be a Banach lattice and Q a positive operator on X. Then the ideal /(Range Q) generated by Range Qis invariant under (Q].

Proof. Let T G (Q\. Since Q is linear,

/(Range Q) = {x : \x\ < o;|Qy| for some y € X and a G [0, oo)}.

So, if x £ /(RangeQ), then |x| ^ a|<5y| for some y E X and a G [0, oo). Thus, since T and Q are positive operators,

\Tx\ ^ T\x\ ^ Ta\Qy\ = aT\Qy\ < aTQ\y\ ^ aQT\y\ = a\Q(T\y\)\.

This implies Tx G /(Range Q) which proves the lemma. •

Definition 3.10. We say that a collection of operators T localizes a set

A C X if for every sequence (xn) in A, there exists a subsequence {xni) and

a sequence Ki G J- such that KiXUi converges to a non-zero vector.

35 In 2005, R. Anisca and V. Troitsky used the method of minimal vectors to prove the following theorem.

Theorem 3.11 ([11])- Suppose that Q is a positive quasinilpotent one-to- one operator with dense range and XQ G X+ with ||xo|| > 1. // the set of all operators dominated by Q localizes B(XQ, 1) fl [0, x0], then there exists an invariant subspace for (Q]. Moreover, if[0,Q] localizes B(xo, 1) fl [0, Xo], then (Q) has an invariant closed ideal.

Theorem 3.11 can be extended as follows.

Theorem 3.12. Suppose that Q is a positive quasinilpotent operator on a

Banach lattice X and x0 G X+ with \\XQ\\ > 1. Then the following are true.

(i) If there exists R in (Q] such that the set of all operators dominated by

R localizes B(xo, 1) fl [0, x0], then there exists an invariant subspace for {Q}-

(ii) If [0,R] localizes B(x0,1) fl [0, xo] for some R in (Q], then (Q] has an invariant closed ideal.

Proof, (i) Suppose that that the set of all operators dominated by R localizes

B(xo, l)n[0, x0]. We show that there exists a common nontrivial invariant sub- space for (Q). We assume /(Range Q) is dense in X, otherwise by Lemma 3.9, /(RangeQ) is a nontrivial closed invariant ideal for (Q]. Fix e > 0, for every n ^ 1, choose a (1 + e)-minimal vector yn and a minimal functional fn for n Q and B(x0, r) + X+. By Lemma 3.8, there is a subsequence (yni) such that

H ~ii —• 0. Since ||/ni|| = 1 for all i, we can assume (by passing to a further subsequence) that (/nJ weak*-converges to some g G X*. By Lemma 3.6(h), we have fn(x0) ^ 1 for all n, it follows that g(xo) ^ 1. In particular, g ^ 0.

Consider the sequence (x0 AQ^^y^-i)^. The terms of this sequence are positive and bounded by XQ, and by Lemma 3.1, they are contained in B(x0,1). ni l So, x0AQ ~ yni-i G B(x0, l)n[0, XQ\. Since the set of all operators dominated by R localizes B(xo, 1) fl [0,x0], by passing to yet a further subsequence if

36 necessary, we find a sequence (Ki) such that Ki is dominated by R for all i ni l and Ki(x0 A Q ~ yni-i) converges to some vector w^O. We claim that g(QTw) = 0 for every T G {Q}. Indeed, suppose T e (Q].

Since fniTQ is a positive functional,

1 fmiQTKiixo AQ"'- ^-!))! ^ /nt(QT|^(x0 AO^-V,-!)!)- (3-1)

Since each fQ is dominated by R,

fnz(QT\Kx{x0 AQ">-VH-I)|) < frHiQTRixoAQ"*-^!)). (3.2)

ni 1 ni l Since fniQTR is a positive functional and xo A Q ~ yni-i ^ Q ~ Vni-i,

ni 1 fni(QTR(x0AQ - yni-l)) ^ fnXQTRiQ^-'yn^)). (3.3)

Since (Q] is a multiplicative semigroup, TR € (Q]. So,

n 1 ni n fni(QTR(Q ^ yn^)) ^ fn%(Q TRyn^) ^ Q* *fni(TRyn^). (3.4)

By Lemma 3.6(vi) and definition of the dual operator,

||Q--/„,|| • IITHII • ll^-B < ('+^ll'ollFfi|IHl^-.ll, (3.5) WVriiW

From the above sequence of equations (3.1), (3.2), (3.3), (3.4), and (3.5), we get fn^QTKiixoAQ^y^)) < (l + g^llgollllTflllll^-!! 112/^1 Thus,

fni(QTKi(x0 AQ^y^)) ^ 0.

On the other hand,

QTKiixo A Qn'" Wi) - QTw

37 in norm. Since fn. ^-* g, we conclude that g(QTw) = 0, hence (Q*g)(Tw) = 0. Since the ideal generated by Range Q is dense and g ^ 0 is positive, we have Q*g ^ 0. Let Y be the linear span of (Q]w, that is, Y = lm{Tw : T G (Q}}. Since (Q] is a multiplicative semigroup, Y is invariant under every T G (Q]. It follows from 0 ^ w £ Y that y is non-zero. Finally, Y ^ X because Q*g vanishes on Y. This proves (i).

(ii) Suppose that [0,R] localizes B(xo,l) D [0, XQ] for some xQ ^ 0 and ||:co|| > 1- Then the vector w constructed in the previous argument is positive. Let E be the ideal generated by (Q]w, that is

E = {y £ X | \y\ ^ Tw for some T G (Q]}.

The ideal E is non-trivial since the identity operator is in (Q] so that w G E, and E is invariant under (Q], indeed, if S G [Q] and x £ E (i.e., |x| ^ Tw for some T G (

Recall that an operator on a Banach lattice is AM-compact if it maps order bounded sets to relatively compact sets. In [27], the authors proved the following theorem which is an extension of earlier results by R. Drnovsek (see [1, Theorems 10.44 and 10.50]):

Theorem 3.13. If Q is a quasinilpotent positive operator on a Banach lattice with a quasiinterior point such that some operator in [Q) dominates a non-zero AM-compact operator, then [Q) has an invariant closed ideal.

Our next theorem provides a similar result for (Q].

Theorem 3.14. If Q is a positive quasinilpotent operator on a Banach lattice and there exists a non-zero AM-compact operator K dominated by an operator in (Q] then (Q] has an invariant subspace. Furthermore, if K ^ 0 then (Q] has a closed invariant ideal.

38 Proof. Let K : X —> X be a non-zero AM-compact operator dominated by an operator R G (Q]- Pick u ^ 0 such that XM ^ 0. We can do so because otherwise, if Ku = 0 for every u ^ 0, then ifrc = /C:r+ — ifx- = 0 for every igl which is a contradiction to the assumption that K ^ 0. Define x0 = ||£}u. Notice that ||x0|| > 1 and \\Kx0\\ = 2\\K\\.

Claim: 0 ^ K(B(x0,1) fl [0, a;0]). Indeed, suppose there exists a sequence

(xn) in B(x0,1) fl [0, x0] such that Kxn —> 0.

ll-RTxoll - H-ftT^ - Kx0\\ ^

||Kx0||-||K||||xn-x0||^||Kxo||-||/r||.

This implies ||ifxo|| ^ \\K\\ + H-K'Znll for every n. Letting n —»• oo, we get ||-K"xo|| ^ 11-^11 which is a contradiction.

Since if is AM-compact, for every sequence xn'£ J5(x0,1) fl [0, xo], there exists a subsequence (xni) such that X:rni converges. Therefore, the set of operators dominated by R localizes B(xo, 1) fl [0,xo]. Thus, Theorem 3.12 completes the proof. •

Since every compact operator is an AM-compact operator, we have the following simple consequence of Theorem 3.14.

Corollary 3.15. If Q is a positive quasinilpotent operator and there exists a non-zero compact operator K dominated by an operator in (Q] then {Q] has an invariant subspace. Furthermore, if K ^ 0 then (Q] has a closed invariant ideal.

3.3 Invariant subspaces of compact-friendly op­ erators

Recall that a positive operator Q : X —> X is said to be compact-friendly if there are two positive operators R, K and a non-zero operator C such that

39 RQ = QR, K is compact and

\Cx\ ^ R\x\ and \Cx\ < K\x\ for each x.

Remark 3.16. If Q is a quasinilpotent compact-friendly operator and C3 ^ 0 where C is as in the definition of compact-friendly operators then (Q] has a common invariant subspace. Indeed, by Theorem 2.11, C3 is compact and C3 is dominated by R3 which is in (Q}. Then we use Corollary 3.15. Furthermore, if C is positive then by Theorem 3.12 and Corollary 3.15, (Q] has a common invariant ideal.

The preceding remark shows that, for a quasinilpotent compact-friendly operator, the question of the invariant subspace problem has an affirmative answer when C3 7^ 0, where C is as in the definition of the compact-friendly operator. In fact, we will show that the question has also an affirmative answer in the case that C3 = 0. Several results have been established regarding the invariant subspace problem for compact-friendly operator. For instance, Y. A. Abramovich, C. D. Aliprantis, and O. Burkinshaw proved the following theorem by using results of R. Drnovsek (Theorem 2.21).

Theorem 3.17 ([1]). If a non-zero compact-friendly operator Q : X —> X on a Banach latttice is quasinilpotent at some XQ > 0, then Q has a non-trivial closed invariant ideal.

Moreover, for each sequence {Tn} in [Q) there exists a non-trivial closed ideal that is invariant under Q and under each Tn.

Some additional assumption on a compact-friendly operator T are neces­ sary if one wants to establish the existence of a non-trivial invariant closed ideal for the collection of all positive operators commuting with T. Indeed, the identity operator is compact-friendly but does not have any non-trivial closed invariant ideal which is also invariant under each positive operator.

Corollary 3.18 ([1]). Let Q : X —> X be a positive operator on a Banach lattice and suppose that (Q] contains a non-zero compact-friendly operator that

40 is qiiasinilpotent at some non-zero positive vector. Then Q has a non-trivial closed invariant ideal.

For Dedekind complete Banach lattices, Theorem 3.17 can be improved by showing the existence of a common non-trivial closed invariant ideal for [Q).

Theorem 3.19 ([1]). // a non-zero compact-friendly operator Q : X —» X on a Dedekind complete Banach lattice is quasinilpotent at a non-zero positive vector, then [Q) has a non-trivial closed invariant ideal.

The preceding two theorems showed the existence of invariant subspaces for subsets of the super-right commutant of a positive operator Q under the assumption stated. The next two theorems show that under similar assump­ tion, the super-left commutant (Q] has an invariant subspace. The proofs are similar to the proofs of Theorems 3.17 and Theorem 3.19 (see [1]), but we use Corollary 3.15 instead of Drnovsek's theorem (Theorem 2.21) as we deal with [Q] instead of [Q).

Theorem 3.20. If Q is a quasinilpotent compact-friendly operator then at least one of the following is true:

(i) for each sequence {Tn} in (Q] there exists a non-trivial closed ideal that

is invariant under Q and each Tn, or

(ii) (Q] has an invariant subspace.

Proof Without loss of generality we can assume that ||Q|| < 1 and suppose that (Tn) is a sequence in (Q]. Pick arbitrary scalars an > 0 that are small a enough so that the positive operator T = J^li nTn exists and \\Q + T\\ < 1. Since (Q] is a norm closed additive semigroup, it follows that the positive operator A = Yl^LoiQ + ^)" belongs to (Q}.

For each x > 0 we denote by Jx the principal ideal generated by Ax;

Jx = {y <= X : \y\ ^ \Ax for some A > 0}.

It follows from x ^ Ax that x G Jx, so that Jx ^ 0. 41 Observe that Jx is (Q + T)-invariant. Since 0 ^ Q,T ^ Q + T, Jx is invariant under Q and T and thus it is also Tn-invariant for each n. Therefore, if Jx 7^ X for some x > 0 then Jx is the desired invariant ideal.

Suppose Jx = X for each x > O.In other words, Ax is a quasi-interior in X for each x > 0. Fix two positive operators i?, K and an operator C such that RQ = Q-R, K is compact and

|Cx| ^ R\x\ and |Cx| ^ K\x\ for each x.

Since C^O, there exists some Xi > 0 such that Cx\ ^ 0. Then A|Cxi| is a quasi-interior point satisfying J4|CXI | ^ \Cx\ |, and it follows from Lemma 2.10, that there exists an operator Mi : X —• X dominated by the identity operator such that x2 = M\Cx\ > 0. Put Si = M\C. Observe that S\ is dominated by K and R.

Since JX2 = X and C 7^ 0, there exists 0 < y < Ar2 such that Cy 7^ 0. Hence A|Cy| is quasi-interior point. Since \Cy\ ^ A|C?/|, by Lemma2.10 there exists M2 : X —> X dominated by the identity operator such that

£3 = M2Cy > 0. Since \y\ ^ Ax2 and Ax2 is quasi-interior, it follows from Lemma2.10 that there exists M : X —> X dominated by the identity operator such that MAx2 = y. So x3 = M2Cy = M2CMAx2. Let 52 = M2CMA and note that S2 is dominated by the compact positive operator KA and by the operator RA.

If we repeat the preceding arguments with the vector x2 replaced by X3, then we obtain one more operator S3 : X —* X that satisfies 53X3 > 0 and dominated by KA and RA. Consider the operator S3S2S1. From S3S2S1X1 = S3X3 > 0, we know that S3S2Si =£ 0 Moreover, as shown above, each operator Si (i=l, 2, 3) is dominated by a compact operator and, therefore, Theorem 2.11 guarantees that S3S2S1 is compact. Observe that

{SsS^xl ^ RARAR\x\

42 for each x E X. Let S = RARAR. Clearly S e (Q]. Then Corollary 3.15 guarantees that (Q] has a common invariant subspace. D

Remark 3.21. If X is order complete then we may assume that the operator C in the definition of compact-friendly operator is positive. Indeed, take x ^ 0. For each y G [—x, x], we have \Cy\ ^ K\y\ ^ Kx. Thus, {\Cy\ : \y\ ^ x} is a bounded set. Since X is order complete, sup{|C?/| : \y\ ^ x} exists. Then, by Theorem 2.9, the modulus of C exists and \C\x = s\xpyerx>xi\Cy\ ^ Kx, so that \C\ ^ K. Likewise, \C\ ^ R. So, in the above definition, if C is not positive, we can pick the modulus \C\ of C.

Theorem 3.22. If a non-zero compact-friendly operator Q on an order com­ plete Banach lattice is quasinilpotent, then (Q\ has a non-trivial closed invari­ ant ideal.

Proof. For each x > 0 we denote by Jx the ideal generated by the orbit {Q], that is,

Jx = [y e X | \y\ < Tx for some T G (Q]}.

Since x G Jx, we have Jx ^ 0. Note that Jx is invariant under each T G (Q\.

Therefore, if Jx ^ X for some x > 0, then Jx is a (Q]-invariant closed ideal.

So, suppose Jx = X for each x > 0. By Remark 3.21, there exist three positive non-zero operators R, K and C such that RQ — QR, C ^ R, C ^ K, and K is compact. Claim: For every x > 0, there exists A G (Q] such that CAx > 0. Indeed, since Jx = X and C^0, there exists a positive y 0. Then y ^ A:r for some A G (Q], hence CAx > 0.

Fix any x > 0. Applying the claim three times, we find Ai:A2,A3 G (Q] such that CA3CA2CA1x > 0. Let S = Ci43CA2Cyli. Then 5 ^ 0 and CAi < KAi (i = 1,2,3), hence S is compact by Theorem 2.11. Also, 0 ^ S < RA3RA2RA1 G (Q]. Then Corollary 3.15 guarantees that (Q] has a non-trivial closed invariant ideal. •

43 Chapter 4

Invariant subspaces of operators on ordered Banach spaces

In this chapter, we produce invariant subspaces for some operators defined on ordered Banach spaces. We extend the method of minimal vectors to or­ dered Banach spaces. In the sequel, we use the method to produce invariant subspaces of operators on ordered Banach spaces, such as Sobolev spaces. First, we introduce some notations and terminologies related to ordered Ba­ nach space.

4.1 Ordered Banach spaces

Recall that a vector space X is called an ordered Banach space if it is a

Banach space and an ordered vector space such that the positive cone X+ is norm-closed. If, in addition, X = X+ — X+, then we say that X is a Banach space with a generating closed cone. A lattice-ordered Banach space is an ordered Banach space with the order being a lattice order. Note that a lattice-ordered Banach space X is a Banach lattice if \x\ ^ \y\ implies ||a:|| ^ \\y\\ for all x,y e X.

A few examples can be mentioned. The classical spaces Lp(p,), where /i is a measure and 1 ^ p ^ oo, as well as CQ(Q), where Q is a locally compact

44 Hausdorff space, are Banach lattices. However, there are many important ordered Banach spaces with generating cones which are not Banach lattices. One could mention, in particular, the spaces Ck[0,1] of all k times continu­ ously differentiable functions on [0,1], as well as C*-algebras. Sobolev spaces Wk'p(tt) are ordered Banach spaces. Moreover, we observe in Section 4.5 that for k = 1 they are lattice-ordered Banach spaces. However, they fail to be

Banach lattices as the norm is not monotone on X+. Throughout this chapter, X will usually stand for an ordered vector space or an ordered Banach space. For two operators S and T in L(X), we write T ^ S if S — T^O, i.e., S — T is a positive operator. We define some notions in ordered vector spaces.

Definition 4.1. Let X be an ordered vector space.

(i) If A and B are two subsets of X, we say that A minorizes B if for each b in B, there exists a £ A such that a ^ b.

(ii) If S and T are two linear operators on X, we say that T is dominated by S if ±Tx ^ Sx whenever x ^ 0.

Remark 4.2. If X is a vector lattice, Definition 4.1(h) is equivalent to the usual definition that T is dominated by S if \Tx\ ^ 51x1 for every x £ X. Indeed, if \Tx\ ^ -S^l for every x E X, then, for each x ^ 0, we have

±Tx ^ \Tx\ ^ S\x\ = Sx.

Conversely, if ±Tx ^ Sx for all x ^ 0, then, for every x € X, we have

±Tx = ±{Tx+ - Tx~) = ±Tx+ - ±Tx~ ^ Sx+ + Sx" = S\x\.

This implies \Tx\ = (-Tx) V (TV) < S\x\.

Suppose X is an ordered Banach space. For A,BC.X and u E X we write u + B = {u + b\b£B} and A + B = {a + b\aeA, be B}. Recall

45 that for S and T in L(X), T ^ S if Tx > Sx for each x > 0. So, if T < 5 in L(X), we write [T, 5] = {R e L(X) | T < i2 ^ 5} (in particular, all the operators in [T,S] are bounded). The commutant of T e £(X) is the set {T}' = {5 e L(X) | T5 = ST}.

Proposition 4.3. Suppose X is an ordered Banach space. Then,

(i) for any a,b £ X, [a, b] = (a + X+) n (6 - X+)

(ii) order intervals are closed sets.

(Hi) L(X) is an ordered Banach space.

Proof. First, we prove (i), then (ii) follows immediately. Let x € [a,b], then x — a ^ 0 and b — x ^ 0. So, writing x = a + x — a and x = b — (b — x) shows that x G (a + X+) (~)(b- X+). On the other hand, if x £ (a + X+) D (b - X+), m then there exist x\ and £2 X+ such that x = a + X\ and x = b — xi- Thus, x ^ a and x ^ b which shows that x 6 [a, 6]. (iii) Obviously, L(X) is a Banach space arid an ordered vector space. We show its positive cone is closed. Indeed, Let (Rn) be a sequence of positive operators such that Rn —> R. Thus, for each x ^ 0, RnX —> i?x. Since i?nx ^ 0 and X+ is closed, we get Rx ^ 0. So, the positive cone of L(X) is closed. Therefore, L(X) is an ordered Banach space. D

Following [1], we define the super left-commutant (Q\ and the super right-commutant [Q) of Q (in the same way as in the case when X is a Banach lattice) as follows:

(Q] = {T > 0 I TQ < QT} and [Q) = {T ^ 0 | TQ ^ QT}.

We will make use of the following well-known theorem (see, e.g., [2]).

Theorem 4.4 (Lozanovski). Every positive linear operator between ordered Banach spaces with generating closed cones is bounded.

46 Recall that a subspace E of a vector lattice X is said to be an (order) ideal if a G E and |x| ^ \a\ imply x G E. If A C X, then 1(A) stands for the smallest ideal in X containing A. It is easy to see that

n a 1(A) = {x G X : \x\ < 2_j ^i\ i\> -^l) • • • i ^7i £ K+ and fli,...,anei}. i=i

We now extend this concept to ordered vector spaces.

Definition 4.5. A subspace E in an ordered vector space is said to be an (order) ideal if

(i) x G E implies that there exists a positive a E E such that x ^ a, and

(ii) ±x ^ a G E implies x G E.

Remark 4.6. Note that, in a vector lattice, this definition agrees with the usual one. Indeed, suppose E is a subspace of a vector lattice such that (i) and (ii) are satisfied. Let \y\ ^ |x| for some x G E. Since x and — x are in E, by (i), we get two positive elements a and b in E such that x ^ a and — x ^ b. This implies ±x ^ a + b G E. Thus,

\y\^ \x\ = (x) V (-x) ^a + b.

This implies ±y ^ a + b G £?. So, by (ii), y £ E. On the other hand, if |x| ^ |y| for some y E E implies x G E, then (ii) is satisfied. In addition, since, for each x G E, x ^ |x| and |x| G E, (i) is satisfied. The relevance of the preceding definition of ideals in ordered vector spaces is well demonstrated in the following example.

Example 4.7. Let P[0,1] be the space of all polynomials on [0,1]. Consider the point-wise order in P[0,1], i.e, / ^ g if f(t) ^ g(t) for each t G [0,1]. Note that this order is not a lattice order. Consider the vector subspace E= {/ G P[0,1] : /(0) = 0} of P[0,1]. Clearly, if ±f ^ g G E, then / G E. Moreover, if / G E, then the derivative /' of / is also a polynomial. Put

47 M = supfer0)1i f'(t). Now consider the polynomial g(t) = Mt. It is easy to see that g G E+ and / ^ g. So, E is an ideal in the ordered vector space P[0,1].

If X is an ordered vector space and A n X+ =fi 0, we will write

Io{A) = {x G X | ±x < Aa for some A G R+ and some a E A}.

Note that Io(A) need not contain A.

Lemma 4.8. Suppose that X is an ordered vector space.

(i) If A is a convex subset of X then IQ(A) is an ideal; Io(A) is contained in every ideal containing A.

(ii) A subset E C X is an ideal if and only if E = \Ja&E Io(a).

(Hi) Suppose that E and F are two ideals in X such that Io{a) H Io(b) is an ideal whenever O^aeB and 0 ^ b G F. Then E D F is an ideal.

Proof, (i) Suppose A is a convex subset of X. Let x, y G Io(A). Then ±x < \±a and ±y < X2b for some Ai, A2 G M+ and a, 6 G A This implies

2 ±(x + y)^ \ia + X2b = (Ax + A2)( \ a + 6). ^1 + ^2 -"M + ^2

a G By convexity of A, we have x[TX^ + A7+A^ ^" So, x + ?/ G /0(^)- By definition, Io(A) is closed under scalar multiplication. So, Io{A) is a vector subspace. Let x G Io(A). Then ±x ^ Aa for some positive real number A and a G A. x ^ Aa implies —Aa ^ —x. So, —Aa ^ — x ^ Aa. Thus, 2Aa ^ 0. So, for each x, there exists a positive element 6 G io(-<4) such that x ^ b. In addition, if ±y ^ a for some a G Io(A), then x G io(-<4) by definition. This proves Io(A) is an ideal. Moreover, if E is an ideal containing A, then x G /o(-A) implies x £ E. (ii) Suppose i£ is an ideal. Let x G E. Then —x G E. So, there exist a, b G £•+ such that x ^ a and —x ^ 6. Thus, ±x ^ a + b G £+. This implies x G Jo(a + &)• So, E C |Ja6£. /o(a). On the other hand, if x G \JaeE Io(a),

48 then x £ Io(a) for some a £ E. This implies ±x ^ Xa for some A £ R+. So, a x £ E. Therefore, i? = UaeE ^o( )- Conversely, suppose E = \Ja&E Io(a). For x, y £ E, there exist a, b £ E^ and Ai, A2 G M+ such that ±x ^ Aja and

±y ^ X2b. So, ±(x + ?/) < Aia + X2b which implies x + y £ Jo(^ia + X2b). So, x + y £ E. Similarly, Xx £ E for each scalar A and x £ E. Thus, E is a vector subspace. For each x £ E1 there exist a £ E+ such that x £ Io(a), so that x £ Xa for some A £ R+. Moreover, ±x ^ a for some a £ E implies x £ E. So, E is an ideal. (iii) Since £* and F vector subspaces, E n F is also a vector subspace. Suppose x £ E d F. Then, by (ii), there exist a positive vector a £ E and a positive vector b £ F such that x £ Io(a) H /o(^)- Since Io(a) n /o(b) is an ideal by hypothesis, it follows from (ii) that x ^ Ac for some c € Io(a) n /o(^) and A £ R+. Notice that Xc £ Ef) F. Moreover, for any y £ X, if ±y ^ o for some ae£flF, then y £ EnF. •

Lemma 4.9. Suppose that X is a vector lattice and A Q X such that either

(i) A is a convex subset of X+, or

(ii) A is the range of a positive operator on X.

ThenI0{A) = I{A).

Proof, (i) By Lemma 4.8(i), Io(A) is an ideal, and it is the smallest ideal containing A. By Remark 4.6, Io(A) is an ideal in the vector lattice sense.

Thus, I0(A) = 1(A). (ii) Suppose A = Range Q for some positive linear operator Q on X. Since Q is linear, 1(A) = {x £ X : \x\ < Qy for some y £ X}.

Notice that, for x, a £ X, ±x ^ a if and only if |x| ^ a. So,

1(A) = {x £ X : ±x ^ Qy for some y £ X},

which is exactly I0(A). Therefore, IQ(A) = 1(A). •

49 Lemma 4.10. Let X be an ordered vector space with generating cone and Q a positive operator on X. Then Io(R&ngeQ) is an ideal invariant under (Q].

Proof. Since RangeQ is a convex set, by Lemma 4.8(i), Io(Ra,ngeQ) is an ideal. Let T E (Q) and x E I0(R&ngeQ). Thus, T ^ 0 and TQ ^ QT and

±x < Qy for some y E X+. Therefore,

±Tx < TQy < QTy.

This implies Tx E I0(R&ngeQ). So, /0(R-angeQ) is an ideal invariant un­ der (Q\. D

Note that if X is a Banach lattice and E is an ideal in X then E is still an ideal. This remains true if X is a lattice-ordered Banach space with continuous lattice operations. Recall that various lattice operations can be expressed via each other. For example,

x + y + \x-y\ x + y-\x-y\ x\Jy— and x/\y = . y 2 2

So, the continuity of any of the maps x i-* x+, x *—> x~, x i—> |x|, x »-> x A y, and IHJVI/ implies the continuity of the other maps and joint continuity of x V y and x Ay.

Proposition 4.11. Suppose that X is a lattice-ordered Banach space with continuous lattice operations and E is an ideal in X. Then E is again an ideal.

Proof. Suppose that x E E. Take a sequence (xn) in E such that xn —> x.

Then E 3 \xn\ —> |x|, hence |x| E E. Suppose now that 0 ^ y ^ x E E.

Again, take a sequence (xn) in E such that xn —> x. Then, for every n, we have E 3 \xn\ Ay^xAy = y, hence y E E. Suppose \y\ ^ |x| for some x E E. Thus, \x\ E E, and as a result y+, y~ E E. So, y = y+ — y~ E E. D

50 4.2 Positive quasinilpotent operators on Krein spaces

Suppose that X is an ordered Banach space (in particular, X+ is closed).

Recall that u G X+ is said to be a (strong) order unit if for every x G X there exists A > 0 such that ±x ^ \u or, equivalently, if Io(u) = X. We make use of the following standard lemma.

Lemma 4.12 (c.f. [2, Lemma 3.2]). Suppose that X is an ordered Banach space and u G X+. Then the following statements are equivalent:

(i) u is an order unit;

(ii) u G Int(X+);

(in) XB(0,1) C [—u, u] for some A G K+.

Proof. (iii)=>(i) is obvious since, for each x G X, we have

±X^-eXB(0,l)C[-u,u}

u for some A G R+ so that ±x ^ ^ -

To show (ii)=^(iii), suppose that u G \nt{X+). Then u + XB(0,1) C X+ for some A 6 M+. It follows that, for every x G X with ||x|| ^ A, we have u ± x ^ 0, so that —w ^ a; ^ u, hence \B(0,1) C [—u, u]. To show (i)=>(ii), suppose that u is an order unit. Then X = (J^Li [_™> nu]- By Proposition 4.3 (ii), the interval [—nu,nu] is closed for every n. By the Baire Category Theorem, [—u, u] has non-empty interior, so that B(a,X) C [—u, u] for some a G X and A G R+. Now, suppose that x G B(u,X), then

a ± (x — it) G 5(a, A) C [—u, u].

Now, — u ^ a+(x —«) implies x+a ^ 0, while a— (x — u) ^ « implies x — a ^ 0. Adding these two inequalities together yields x ^ 0, so that B(u, A) C X+. D

51 Definition 4.13. An ordered Banach space with an order unit is said to be a Krein space.

Clearly, the positive cone in a Krein space is generating. We would like to mention the following result, which is a corollary of Krein's theorem [34], see also [2, 46, 42].

Theorem 4.14. If T is a positive operator on a Krein space, then T has a closed hyperinvariant subspace.

Suppose that X is an ordered Banach space. Recall that w G X+ is said to be quasi-interior if IQ(W) is dense in X. Clearly, every order unit is quasi-interior.

Lemma 4.15. If X is a Krein space, then every quasi-interior point is an order unit.

Proof. Let w G X+ be quasi-interior. By Lemma 4.12 there exists u G X+ such that 5(0,1) C [-u,u\. It follows that 2^ + 5(0,1) C [u,3u\. Since Io(w) is dense, it meets 2u + 5(0,1). It follows that there exists x £ 5(0,1) such that u ^ 2u + x ^ Aw for some A G M+. Therefore, w ^ ju, hence /oM 2 /o(w) = X. U

Theorem 4.16. Suppose that Q is a positive quasinilpotent operator on a Krein space X. Then (Q] has a non-trivial non-dense invariant ideal.

Proof. Since X is a Krein space, there is an order unit u in X+. Let w = Qu. We claim that w is not quasi-interior. Indeed, otherwise it would be a unit by Lemma 4.15, so that u ^ Aw for some A e M.+ . It follows that Qu ^ ju, so that n n n Q u ^ j^u, hence \ Q u is contained in u + X+. Since XQ is quasinilpotent, n n we have X Q u —> 0. But u + X+ is closed, hence 0 G u + X+, a contradiction.

Let x G X, then ±x ^ Aw for some A G M+, so that ±Qx ^ Aw. It follows that RangeQ C Io(w), so that Io(Ra,ngeQ) = Io(w), hence I0(R&ngeQ) is not dense. By Lemma 4.10, io(Range<3) ^s invariant under each operator in (Q]. So, Io(R&nge Q) is a common non-trivial non-dense invariant ideal for (Q]. •

52 4.3 Applications to C(K) and Ck(Q) spaces and to C*-algebras

In this section we apply the results of the preceding sections to unital uni­ form algebras. We will see that these algebras are Krein spaces, and apply Theorem 4.16 to find invariant closed subspaces for positive quasinilpotent op­ erators on them. Moreover, these subspaces can be chosen to be the closures of non-dense invariant order ideals. It should be pointed out that in this setting one has to distinguish between the algebraic ideals and the order ideals. The non-unital case will be considered in Section 4.7. Clearly, for every compact Hausdorff space K, the space C(K) is a Krein space with 1 being an order unit. Hence, Theorem 4.16 yields the following.

Corollary 4.17. If Q is a positive quasinilpotent operator on C{K) where K is a compact Hausdorff space, then (Q] has a non-trivial closed invariant ideal.

Let £1 be an open connected subset of M.n. Recall that the space Cfc(Q) consists of all real-valued functions / on Q such that Daf is bounded and uniformly continuous on Q whenever |a| ^ k. With the norm defined by 11/11 = S|ai the set Ck(0.) is an ordered Banach space (see [28]). Clearly, 1 is an order unit in Ck(fl), hence Ck(Q) is a Krein space. Therefore, by Theorem 4.16 we have the following.

Corollary 4.18. If Q is a positive quasinilpotent operator on Ck(fl) then (Q] has an invariant non-dense order ideal. In particular, it has a closed invariant subspace.

Before we proceed to C*-algebras, we would like to make a remark about applicability of our techniques to complex Banach spaces. In all our previous results we assumed X to be an ordered Banach space over real scalars. By a complex ordered Banach space we understand the complexification Xc — X + iX of an ordered Banach space X over R (see [46, p. 214]) on complexification procedure for ordered Banach spaces). An operator T £ L(XC) is said to

53 be positive if T(X+) C X+, where X+ is the positive cone of X. Suppose

that X+ is generating in X. Then every positive operator T in L(XC) is the

complexification of its restriction to X, that is, T = (T\x)c- It is easy to see that if a subspace V of X is invariant under S £ L(X), then V + iV is a

subspace of Xc invariant under Sc. Hence, in order to prove that a positive

operator T on Xc has an invariant subspace, it suffices to find an invariant subspace for the restriction of T to X. Let A be a C*-algebra. Then A is a Banach space over C. Observe that

its self-adjoint part Asa is a real Banach space. Recall that for x £ Asa we write x ^ 0 when a(x) C [0, +00). With this order, the positive cone A+ is

closed and generating (generally, Asa is not a lattice). Furthermore, A can be

viewed as the complexification of Asa. Suppose that T is a positive operator

on A. By the preceding paragraph, if its restriction to Asa has an invariant

subspace in Asa, then T has an invariant subspace in A. Hence, it suffices to

look for invariant subspaces of positive operators on Asa.

If, in addition, A is unital, then Asa is a Krein space. Indeed, suppose

that x £ Asa with ||x|| ^ 1. Then a(±x) C [—1,1] and the Spectral Mapping

Theorem yields a(e ± x) C [0,2] C R+, so that ±x ^ e and, therefore,

x £ [—e,e\. It follows from Lemma 4.12 that Asa is a Krein space. Hence, Theorem 4.16 yields the following result.

Theorem 4.19. If Q is a positive quasinilpotent operator on a unital C*- algebra then (Q] has a non-trivial invariant non-dense order ideal. In partic­ ular, it has a closed invariant subspace.

4.4 Applications to Sobolev spaces Wk,p(Q)

Throughout this section, we assume that 0 is a bounded open subset of ~RN, 1 ^ p ^ co, and k £ N. Recall that the Sobolev space Wk'p(Vt) is defined as the set of all those functions / £ LP(Q) for which the weak partial derivatives

Daf exist and are in LP{Q) for each multi-index a with |a| ^ k. The norm on

54 Wk'p(Q) is given by

if p < oo, L|a|a * E IPa/IL ifp = DO, lalsgfc

r where ||-||LP is the norm in Lp(0). As usually, we will write 1 for xn- F° more details on Sobolev spaces we refer the reader to [51, 31, 43, 28]. k p We equip W ' (Q) with the a.e. order, i.e., the order inherited from Lp(f2).

Note that X+ is norm-closed. Indeed, let (/„) be a sequence in X+ such that fc p /„ —> / in W ' (fi). Then ||/n — f\\Lp —> 0, so that / ^ 0 because the positive k,p cone of LP(Q) is closed. Thus, W (tt) is an ordered Banach space. It is k p well known that as a Banach space W ' (Q) is isomorphic to Lp(0,1) when 1 < p < oo, see, e.g., [43, Theorem 11]. However, we will see that the order k,p structure of W (Vt) is very different from that of LP(Q). Remark 4.20. Observe that the norm of Sobolev spaces is generally not mono­ tone. For example, let / e W1'1^, 1] be defined as follows: f(t) = t for all t G [0,1]. Then 0 ^ / ^ 1, but ||/|| > ||1||. If p = oo, then 1 is an order unit in Wk,°°(Q,). It follows that Wk'°°(Q) is a Krein space, so that if Q is a positive quasinilpotent operator on Wk'°°(fl) then (Q] has an invariant non-dense ideal by Theorem 4.16. For this reason, from now on we assume that p < oo.

Let X = Wk'p(Q) such that Q, is regular (i.e., dfl is of class C0,1, see, e.g., [28] and [31] for the precise definition of regular domains) and p > N/k or p = N/k = 1. The classical Sobolev embedding theorem (Theorem 2.2) asserts that, in this case, X continuously embeds into L^fi), i.e., there exists C > 0 such that \\f\\oo < C||/|| for every / e X. It follows that the unit ball of X is contained in [—Cl,Cl]. Combining this observation with Lemma 4.12, we obtain the following result.

Theorem 4.21. If Q is regular and p > N/k or p = N/k = 1 then Wk,p{Q) is a Krein space.

55 In particular, VF1,p[0,1] is a Krein space for any 1 ^ p < oo. The next result now follows immediately from Theorems 4.4, 4.21, 4.14, and 4.16.

Theorem 4.22. Suppose that £1 is regular and p > N/k or p = N/k = 1. If Q is a positive linear operator on Wk'p(Q) then Q is bounded and there is a closed subspace invariant under {Q}'• If, in addition, Q is quasinilpotent, then (Q] has a non-dense invariant ideal.

4.5 A special case: Wl'p{Q)

In this section, we consider the case k = 1. The following fact from [31] shows x that W *($l) is a sublattice of LP(Q).

Lemma 4.23 ([31, Lemma 7.6]). If f E W^(Q) then /+, /~, and \f\ are also in Wl'p{Q) and

/>0 df+ gdi , />0 df~ / = o oXi \ (\ f < 0

for each i = 1,..., N (the equalities are a.e.).

As usual, |/|, /+, and / are defined pointwise. Hence, H/1'P(Q) is a vector lattice and |||/||| = ||/|| for every / G W^P(Q) (recall that W1J,(Q) is generally not a Banach lattice by Remark 4.20). In particular, the positive cone of Wl,p(Vt) is generating. Together with Theorem 4.4 this immediately yields the following.

Corollary 4.24. Every positive linear operator on Wl,p{Vt) is bounded.

Theorem 4.25. The Sobolev space W1,P(Q) is a lattice-ordered Banach space with continuous lattice operations.

Proof. Let X = W1,P(Q). In view of the preceding discussion, we only need to verify the continuity of the lattice operations in X. Take (/n) in X such that

56 /„ - / for some feX.lt follows that \\fn - f\\Lp -* 0 and || §£ - J£ \\Lp - 0

+ for each i. Since I/P(fi) is a Banach lattice, ||/+ — / ||LP —* 0. It is left to show that ||-g^— -g£—1| —>• 0 for every i. Consider the following subsets of Q: A+ = {f > 0}, A' = {/ < 0}, and A0 = {/ = 0}. Also, for every n define - A+ = {fn > 0} and A" = {/„ ^ 0}. In order to show that

df+ or 0, / dxi dxi Jn

+ + we split it into integrals over the following six sets: A n A+, A (1 A°n , A~ n A+, A~ n A%-, A0 n A+, and A0 n 4°-. In fact, by Lemma 4.23, the integrand vanishes a.e. on A" n.A°~ and A0 H A°~, so that there is nothing to do about these two sets. Next, we consider A+DA+ and A°nA+. Lemma 4.23 yields that §^ = ff1 + 0 a.<.e . on At and -4—dxi = -?r-dxi a.e. on A . Further, on A we have -J^- = 0, hence 1 dxiW = ¥- a.e.. Therefore,

df+ or Ofn df_ Ofn df_p € 0. (A+L)A°)nA+dx; dx; I,(A+uA°)nA+ dxi dxi dxi dxi

For the remaining two sets, we show first that the sequences m(A+ r\A^~) and m(A~~ D A+) tend to zero as n —> oo, where m stands for the Lebesgue — measure on fl. Indeed, since \\fn — /||L > 0, by passing to a subsequence we + may assume that fn —•>• f on fl. In particular, fn —-> f on A . By Egoroff's theorem we can find B C A+ such that /„—>•/ uniformly on 5 and m(B) ^ m(.4+) - e. Since A+ = U£Li(/ > £}> we have m({/ > £}) > m(A+) - £ for some A;. It follows that /„ > 0 on {/ > £} D B for all sufficiently large n, so that J4£- n{/>|}flB = 0. Therefore, m(A+ n A°_) -> 0. Similarly, we show that m(A~ D ,4+) -» 0. Now Lemma 4.23 yields

df_ dfl_dT 0, dxi dxj dxj A+nAl~ A+nA°n-

57 and

dft dP dfn 'A-nA+ dxi dxi A-HA+ dxi dfn df P df < 2P + 2 LA-r\A+ dxj dxj A-HA+ dxi dfn df p df < 2P + 2P 0. dxj dxi A-DA+ dx.

It follows that II^i - ^1II -> 0. • Combining Theorem 4.25 with Proposition 4.11, we immediately obtain the following result.

Corollary 4.26. The closure of every ideal in W1'1'^) is again an ideal.

Y. A. Abramovich, C. D. Aliprantis, and O. Burkinshaw proved the fol­ lowing fact in [5].

Theorem 4.27 ([5]). Let S and T be two positive commuting operators on a Banach lattice such that S is quasinilpotent and dominates a non-zero positive compact operator. Then T has a closed invariant ideal.

A variant of Theorem 4.27 for W1,P(D.) proved in [32] states that an operator on W1'P(Q,) satisfying the same condition has a closed invariant subspace. In fact, this subspace is produced in [32] as the closure of a certain ideal. In view of Corollary 4.26, this closure is itself an ideal. Hence, the following result holds.

Corollary 4.28. Let S andT be two positive commuting operators on Wl,p{Vt) such that S is quasinilpotent and dominates a non-zero positive compact oper­ ator. Then T has a closed invariant ideal.

As in [32], instead of 5 being quasinilpotent, it is sufficient that S be locally quasinilpotent at a positive vector. Theorem 4.22 and Corollary 4.26 together yield the following result.

58 Theorem 4.29. Suppose that 0 is regular and p >Norp = N = l. If Q is a quasinilpotent positive operator on W1,P(Q), then (Q] has a closed invariant ideal.

4.6 Minimal vectors in spaces with a generat­ ing cone

In Chapter 3, we visited the method of minimal vectors on Banach lattices and its application in the context of the Invariant Subspace Problem. This method was also used to produce several generalizations of Theorem 4.27 and of other related results. In this section, we show that the technique developed in Chap­ ter 3 extends naturally to ordered Banach spaces with generating cones. The crucial tool in extending the method is the following well-known theorem (see [2])-

Theorem 4.30 (Krein and Smulian). Let X be a Banach space with a gener­

ating closed cone X+. There exists a constant A > 0 such that for each x £ X

there exist £i,£2 G X+ such that x = X\ — x2 and \\xi\\ ^ A||x||, i = 1,2.

The smallest such A will be denoted A(X). Suppose that Q is a positive bounded operator on an ordered Banach space I,u£ X+, and C G M+. As in Chapter 3, we say that y G X is a C-minimal vector for Q and B(u, 1) + X+ 1 ify^O,y£Q~ (B(u,l) + X+) and \\y\\ ^ Cdist{0, Q^^u, 1) + X+)}.

Lemma 4.31. Suppose that X is a Banach space with a generating closed cone with A = A(X) and Q is a positive linear operator on X such that 7o(Pange Q)

is dense in X. Suppose that u € X+ such that 0 ^ B(u, 1) + X+. Then there exists a (2A)-minimal vector for Q and B(u, 1) + X+.

Proof Note first that Q is bounded by Theorem 4.4. Since io(RangeQ) is dense, it meets B(u, 1). Let x G B(u, 1) n Io(Ran.geQ). Then x < Qh for

some h, so that Qh G B(u,l) + X+, hence Range Q D (B(u,l) + X+) is 1 non-empty. Put D = Q~ (B(u,l) + X+), then D is non-empty. Since Q

59 is continuous, 0 ^ D. It follows that d :— dist{L>, 0} > 0. Find z E D such that ||.z|| ^ 2d. By Theorem 4.30 there exists y G X+ such that z ^ y and

||y|| ^ 2dA. It follows from Qz G B{u, 1) + X+ and Q ^ 0 that Qz ^ Qy, so that Qy G -B(u, 1) + X+, hence y is a (2A)-minimal vector. D

Proposition 4.32. Suppose that Q is a positive bounded operator on an or­

dered Banach space X, u G X+ such that 0 ^ B(u, 1) + X+, and C G R+.

Suppose that y is a C-minimal vector for Q, B(u, 1) +X+, and C. Then there exists f G X* (called a minimal functional) such that

(i) f is positive, and f(u) ^ 1 = ||/||;

(ii) There exists c > 0 such that f\Q(B(o,d)) ^ c and f\B(u,i)+x+ ^ c, where 1 d= dist{0,Q- (B(u,l) + X+)};

(m) illOVIII|y||<(QV)(y)

(v) \\Q*f\\ < ^?-

Proof, (i) and (iii) are proved exactly as the corresponding arguments in Lemma 3.6. The proof of (ii) is exactly the same as the proof of Lemma 3.4(h). Note that C~ly G B{0,d), hence C~lQy G Q(B(0,d)). Since u G B(u,l) + 1 X+, (ii) and (i) imply C~ f(Qy) ^ c ^ f(u) < ||u||, which proves (iv). Finally, combining (iii) and (iv), we get (v). •

For the rest of this section we will make the following assumptions.

(i) X is a Banach space with a generating closed cone;

(ii) u e X+ such that 0 £ B(u, 1) + X+; (4.1) (iii) A C X+ such that A minorizes (B(u, 1) + X+) C\ X+;

(iv) Q is a positive quasinilpotent operator on X.

60 The goal of this section is to show that under these assumptions plus some extra conditions, (Q] has a closed invariant subspace. Before we proceed, we would like to make a few comments on these assumptions.

For A one can always take, e.g., the entire set (B(u, l)+X+)nX+. However, in what follows, we will be localizing A, so that we will be interested in having A as small as possible. If X is a vector lattice, one can always take A =

[u A (B(u, 1) + X+)] fl X+. In [11], where X was a Banach lattice, the set B(u,l)D [0, u] served as A.

Clearly, 0 ^ B(u, 1) + X+ implies ||u|| > 1. The converse is also true if the norm is monotone on X+, i.e., when 0 ^ x ^ y implies ||x|| ^ ||y||. In particular, this holds if X is a Banach lattice. The following example shows that the converse fails in general.

2 Example 4.33. Let X = M ordered so that X+ is the first quadrant. Define a norm on X in such a way that the unit ball is the absolute convex hull of (0,1) and (2, 2). Let u = (0, 2), then ||u|| = 2. However, (-2, 0) G B(u, 1) and (2, 0) G X+, so that (0, 0) G B(u, 1) + X+.

Finally, note that under the assumptions (4.1), Theorem 4.4 guarantees that Q is bounded.

Proposition 4.34. Suppose that X, u, andQ are as in (4.1) and Io(Ra,ngeQ) is dense in X. Put A = A(X). Then for every n e N there exists a positive n (2 A) -minimal vector yn and a minimal functional fn for Q and B(u, 1)+X+. ni Furthermore, there exists a subsequence (rij) such that ,, ~,| > 0 and fni —> g for some non-zero g G X+.

Proof. The existence of sequences of minimal vectors (yn) and minimal func­ tional (/„) follows from Lemma 4.31 and Proposition 4.32. Suppose that there exists 5 > 0 such that jy > 5 for all n, so that \\yi\\ ^ o~\\y2\\ ^ • • • ^ l n

n \\Q yn+1\\ > dist(0,D) ^ M ^ £r\\yn+i\\.

61 It follows that \\Qn\\ i£ f^, which contradicts the quasinilpotence of Q. Hence, n > l( *~ll — 0 for some subsequence (yni)- Since ||/n|| = 1 for every n and the unit ball of X* is weak*-compact, we can assume by passing to a further subsequence that fni ^-> g for some g G X*. Clearly, g ^ 0. Since fn(u) ^ 1 for each n, it follows that g(u) ^ 1, hence g ^ 0. •

Theorem 4.35. Suppose that X, u, A, and Q are as in (4.1). If the set of all operators dominated by Q localizes A then (Q] has an invariant closed subspace. Furthermore, if [0, Q] localizes A then (Q] has a non-dense invariant ideal

Proof. Let A = A(X). By Theorem 4.4, every operator in (Q] and [0,Q] is bounded. In view of Lemma 4.10, we may assume without loss of generality that I0(R&ngeQ) is dense in X. Let (yni), (fni), and g be as in Proposi­ tion 4.34. Then Q^^y^-i G (B(u, 1) + X+) (~1 X+ for each i. It follows that ni l there exists eij G A such that a; ^ Q ~ yni-\. By passing to a further sub­ sequence, we can find a sequence (Ki) such that Q dominates Ki for every i ni and Kiai converges to some w^O. Then ±1^^ ^ Qai ^ Q yni-\. For every T G (Q] we have

± fni(TKiai) < fni{Q^Tyn^) = (Q^fmKTy^) < iiQ^/n.iiimiii^-iii < 4A-j-2lldyl imiiiy^u 112/n* 11 by Propositions 4.32(v). It follows from ^^ -• 0 that fn^TK^) -> 0. On the other hand, fni(TKia,) -> #(Tw). Thus, p(Tw) = 0 for every T G (

62 as w E E. Finally, g ^ 0 implies E C ker#, hence E ^ X. D

Theorem 4.36. Suppose that X, u, A, andQ are as in (4.1), and J0(Range Q) is dense. Suppose also that J- is a family of positive contractive operators on X such that T localizes A and & C L(X) is a multiplicative semigroup such that

Proof. Again, let (yni), (fni), and g be as in Proposition 4.34 with A = A(X). ni l Since Q ~ yni-\ E (B(u, 1) + X+) n X+ for every i, we can find a* E A such ni l that <2j ^ Q ~ yni-\. By passing to a further subsequence, we can find a sequence (Ki) in T such that K^ converges to some w ^ 0. Let T E &.

Since TK{ E (Q], Propositions 4.32(v) yields

ni 0 ^ fni(QTKiai) < fni{Q TKiVn%_x) 2 < iiQ^/n.iiimin^-iii < ^-Vii^iiii^-ii4A lldl i - o \\Vni\\ since ^Y ~^ °' Jt follows that 9(QTw) = ° for every T E B. Let Y be the linear span of &w; then Y is invariant under &. Since Jo(RangeQ) is dense and j^Owe have Q*g ^ 0. This yields Y ^ X because Tw E ker Q*g for every T E 6. Finally, if V = {0}, then Tw = 0 for each T E ©, so that span it; is a one-dimensional subspace invariant under 6. Suppose now that 6 consists of positive operators. Then w can be chosen to be positive. Let A be the convex hull of &w and put E = ioC^); then E is an ideal by Lemma 4.8(i) and E is invariant under &. It follows from Q*g ^ 0 that Q*g vanishes on E, hence E ^ X. If E ^ {0}, we are done. Otherwise, Tw = 0 for each T E &. Put F = IQ(W); then F is a non-trivial ideal; F is invariant under 6 since every T E 6 vanishes on F. Also, F ^ X as in this case every operator in 6 is zero. •

Corollary 4.37. Suppose that X, u, A, and Q are as in (4.1). If the set of all contractions in (Q] localizes A then (Q] has a non-dense invariant ideal.

63 Proof. By Lemma 4.10 we may assume that I0(Range Q) is dense. Now apply Theorem 4.36 with F = {K G {Q] \ \\K\\ ^ l} and 6 = (Qj. D

Remark 4.38. In view of Proposition 4.11, if X is a lattice-ordered Banach space with continuous operations then the invariant ideals in Theorems 4.35 and 4.36 and Corollary 4.37 can be taken to be closed.

4.7 Applications of minimal vector technique

4.7.1 Non-unital uniform algebras

Consider Asa for a non-unital C*-algebra A. An important special case is the space Co(fi) of continuous functions on a locally compact Hausdorff space which vanish at infinity; equipped with sup-norm. These spaces are ordered Banach spaces with closed generating cones (but not Krein spaces). Hence, Theorems 4.35 and 4.36, as well as Corollary 4.37, guarantee that if Q is a positive quasinilpotent operator on any of these spaces satisfying the conditions described in the theorems, then (Q\ has a common invariant closed subspace. Observe that for every u G A+ with ||w|| > 1 we automatically have 0 ^ B(u, 1) + A+ (this is one of the assumptions in (4.1)). Indeed, suppose that

0 G B(u, 1) + A+, then there exist sequences (xn) in 5(0,1) and hn G A+ such that u + xn + hn —> 0. It follows that ||u + hn\\ — \\xn\\ —> 0. However, in a C*-algebra, 0 ^ a ^ b implies ||a|| ^ ||6||; see, e.g., [46, VI.3.2.U]. Hence IIw + ^n|| — Ilxn|| ^ IMI — 1 > 0, a contradiction.

4.7.2 Sobolev spaces

Finally, we present an application of the localization technique developed in Section 4.6 to W1,p(fl). In the following theorem, we do not require that Q is regular or that p and N are related.

64 Being a lattice-ordered space, jy1,p(Q) clearly has a generating cone. Com­ bining Corollaries 4.24 and 4.26 with Theorem 4.35 and Corollary 4.37, we obtain the following result.

Theorem 4.39. Let X = W1,P(D,) and suppose that Q is a positive quasinil-

potent operator on X, u G X+ with 0 ^ B(u, 1) + X+, and A C X+ such that

A minorizes {B(u, 1) + X+) fl X+.

(i) If the set of all operators dominated by Q localizes A then (Q] has an invariant closed subspace.

(ii) if either [0, Q] or the set of all contractions in (Q] localizes A, then (Q] has an invariant closed ideal.

Next, we show that the assumptions in Theorem 4.39 are not as restrictive as they might seem.

k p Lemma 4.40. If X = W > {Q) and u G X+ with \\u\\Lp > 1 then 0 £

B(u,l) + X+.

Proof. Suppose not. Then there exist sequences (xn) in the unit ball of X and

(hn) in X+ such that u + xn + hn —> 0. This implies ||tt + xn + hn\\Lp —> 0, so that Hii + ZinlUp - \\xn\\Lp -^ 0. However, ||« + MLP - \\xn\\Lp ^ ||w||Lp - 1 > 0, a contradiction. •

Remark 4.41. Let X be a lattice ordered Banach space such that |||x||| = ||x|| 1,P for every x G X (for example, X = W (Q)). Let u G X+ such that 0 ^ B(u,l)+X+. Then, of course, ||u|| > 1. Let A = {y+ \ y G B{u,l)}. We

claim that then A is norm-bounded, A C (B(u, 1) + X+) f) X+ so that 0 ^ A,

and A minorizes (B(u, 1) + X+) n X+. Indeed, suppose that y G B(u, 1). We have y+ = ^^, hence \\y+\\ ^ so + \\y\\ ^ IMI + 1) ^at A is bounded. Also, y = y + y~ G B(u, 1) + X+, so that A C (£(u, 1) + X+) n X+. Now, let z G (B(u, 1) + X+) n X+. Then 2 = y + h for some ?/ G B(u, 1) and /i ^ 0. It follows from z ^ 0 and z ^ y + that z > y , hence vl minorizes (B(u, 1) + X+) f) X+.

65 We now show that Theorem 4.39 implies an analogue of Theorem 4.27 for

Theorem 4.42. Suppose that Q is a regular domain and Q is a positive quasi- nilpotent operator on X — Wl'p(£l) and 0 < K ^ Q for a non-zero compact operator K. Then (Q] has a closed invariant ideal.

Proof. Since K ^ 0 we have Kl^ 0 as, otherwise, K would vanish on all bounded functions, but bounded functions are dense in X. It follows that there exists e > 0 such that 0 £ K(B(1, e)). Let u = Jl, then 0 <£ K(B(u,l)).

By scaling u even further we may also assume that \\U\\LP > 1, hence 0 ^

B(u, 1) + X+ by Lemma 4.40. As in Remark 4.41, take A = {y+ \ y G B(u, 1)}; then A minorizes

(B(u, 1) + X+) D X+. Since u is a multiple of 1, it follows from Lemma 4.23 that \\y+ — u\\ ^ \\y — u\\ ^ 1 for every y G B(u, 1); hence A C B(u, 1).

Observe that [0, Q] localizes A. Indeed, let (xn) be a sequence in A. Since sucn K G [0, Q] and K is compact, there is a subsequence (xni) that Kxni —> w for some w. Clearly, w G i^(-A) Q K(B(u,l)), hence w ^ 0. Now apply Theorem 4.39. •

66 Chapter 5

Banach lattice martingale spaces

Consider a probability space (fi,.P, P) and a filtration {^Fn)^Li, i-e., an in­ creasing sequence of sub-sigma-algebras of T (it is often convenient to assume we wr e that T = VnLi 3~n, as otherwise one can replace T with VnLi •?"«)> ^

LP(P) for Lp{tt,T,P).

A sequence A = (xn)^=1 of functions in L\(P) is called a martingale rela­ r tive to (J-n) and P if E(xm\J n) = xn whenever m ^ n, and a submartingale if P(xm|jFn) ^ xn whenever m ^ n. A martingale A = (xn) is Lp-bounded if its Lp-martingale norm, given by ||^4||p = sup„||ar„||p, is finite. Let the symbol Mp = MP(Q,.F, (J-n),P) denote the space of all Lp-bounded martin­ gales.

5.1 Martingales in Banach lattices

R. Douglas proved in [23] (see also [3]) that the conditional expectations are the only contractive projections on LX{P) preserving constant functions. In view of this result of R. Douglas, V. Troitsky introduced a generalization of the concept of martingales in a Banach lattice (see [48]) by replacing the conditional expectations with positive contractive projections. The general

67 definition of a filtration is given as follow.

Definition 5.1. A sequence of positive contractive projections (En) on a Ba- nach lattice X is called a filtration if EnEm = EnAm.

J Note that the ranges of En s form a nested increasing sequence. Here En's play the role of conditional expectations. A filtration is said to be dense if

Enx —> x in norm for every x £ X, or, equivalently, if \J^=1 Range En is dense in X. This is analogous to the condition T = V^Li ?~n f°r classical nitrations. But, unlike in the case of classical filtration, generally we cannot simply replace

X with U^Li Range En because the latter set need not be a Banach sublattice oiX. Note, however, that if a projection P is strictly positive, i.e., Px > 0 whenever x > 0, then Range P is a sublattice of X (see [45, 17]). Therefore, if

Em is strictly positive for some m then (J^Li Range .E^ is a Banach sublattice of X. Indeed, for every x > 0 and every n ^ m, we have Enx ^ 0 because

0 < Emx = EmEnx. It follows that En is strictly positive for all n ^ m, so that

Range En is a sublattice. Thus, when a filtration contains a strictly positive projection, one can assume that it is dense by replacing X with IJ^ Range En.

Definition 5.2. Let X be a Banach lattice.

(i) A sequence A = (xn) of elements of X is called a martingale relative

to a filtration (En) if Enxm = xn whenever n ^ m.

(ii) A sequence A = (xn) is called a submartingale if Enxm ^ xn whenever n ^ m.

(Hi) A (sub)martingale A = (xn) is said to be bounded if its martingale norm given by \\A\\ = supj|x„|| is finite.

Remark 5.3. If A = (xn) is a martingale or a positive submartingale, then the sequence ||xn|| is increasing. Indeed, ||x„|| ^ HE'nXn+iH < ||xn+i||. Therefore,

||J4|| = limn||xn||.

68 The class of all bounded martingales is denoted by M(X, (En)). It is easy to see that M[X, (En)) is a closed subspace of the space (O^Lj X) , hence

M(X, (En)) is a Banach space.

5.2 Examples of martingales in Banach lat­ tices

In the case of classical martingales, we have X = Li(P) and En = E(- \

J-n). Any classical filtration satisfies an important additional property: it preserves norms of positive vectors, that is, H-En^H = ||^|| for every n and every x G X+. V. Troitsky ([48]) presented the following examples of nitrations and mar­ tingales which are related to bases in Banach spaces.

Example 5.4. Suppose that (ei) is a 1-unconditional basis in X such that a e Yl^Li i i ^ 0 if and only if c^ ^ 0 for all i ^ 1. For every n ^ 1, let En a e be the n-th basis projection given by En{^°^x o^e*) = Y^=i i ii then (En) is a dense filtration. Notice that (xn) is a martingale if and only if there a e r exists a sequence of scalars (ctj) such that xn = Y^=i i i f° every n ^ 1. a e A martingale (xn) is convergent if and only if there exists z = ]T^i i i such a e m case x that xn = X^iLi i ii this n converges to z. The basis is boundedly complete if and only if every bounded martingale converges.

Example 5.5. Consider the special case of Example 5.4 when X — c0. Again,

A = (xn) is a martingale in CQ if and only if there exists a sequence of scalars ai6i r everv (Q:J) such that xn = XXa f° n ^ 1. Notice that X is bounded if and only if the sequence (ai) is bounded, and in this case \\A\\ = supj|aij|.

Thus, in this case M(X, {En)) can be identified with i^. Observe that here

M(X, (En)) is non-separable even though X is separable.

Example 5.6. Let X = C[0,1] and (e^) be the Schauder system in C[0,1].

Again, let En be the n-th basis projection. One can easily see that for / 6

69 C[0,1], its image Enf agrees with / on a set of dyadic points and is linear between those points. In particular, every En is a positive operator. Since the Schauder system is a monotone basis, each En is a contraction. Hence,

(En) is a dense filtration on X. Clearly, not every martingale is convergent.

For example, put xn(0) equal 0 at 0, equal 1 at all the other dyadic points corresponding to En, and linear in between. Then (xn) is a non-convergent martingale.

Example 5.7. Let X = Lp[0, +oo) (1 ^ p < +oo) and put Enx = x • X[o,n], i.e., En "cuts-off" the tail of re after n. One can easily see that (En) is a dense filtration. A sequence (xn) is a martingale if xn = Y^=i hi where (hi) is a sequence in X such that supp/ij C [i — l,i\. It can be easily verified that the map x € X —> (x • X[o,n]) is an isometry from X onto M.

5.3 When is M(X, (En)) a Banach lattice?

In [48], an order relation is introduced on (0^! X) as follows: if A = (xn) and B = (yn), we say that A ^ B if xn ^ yn for each n. With this order,

M(X,(En)) is an ordered Banach space. Clearly, the norm is monotone: if 0 ^ A < B then \\A\\ < ||5||. Notice that if the filtration preserves the norms of positive vectors and

A = (xn) is a positive martingale then ||xn|| = H^Xmll = ||xm|| whenever n ^ m, so that \\xn\\ — ^A\\ for every n. Thus, in this case, every positive martingale is bounded.

Later, we will show that, in some cases, M(X, (En)) is not a vector lat­ tice in the order we just introduced. However, under certain conditions,

M(X, (En)) is a Banach lattice. One may ask how we could compute AV B,

A A B, and \A\ for two martingales A = (xn) and B = (yn) in M(X, (En)).

The "natual guess" that AvB = (xnVyn)™=1, AAB = (xnAyn)™=1, and \X\ = x {\ n\) _, turns out to be wrong since (|xm|) is not necessarily a martingale for a 7 martingale (xn). For example, let (Q, J , P) be the unit segment endowed with

Lebesgue measure and (J-n) be the dyadic filtration. Consider an Li-bounded

70 — r martingale defined in the following way: X\ = 0, xn = X[o,i/2) X[i/2,i] f°

n > 1. It is easy to see that (\xn\) is not a martingale. Notice, however, that if

(xn) and (yn) are two submartingales then (xnVyn) is a submartingale. Indeed,

if n ^ m then £ra(xn V r/„) ^ (£m£n) V (Emyn) ^ (xm V ym). In particular, if

(xn) is a martingale, then (\xn\) is a submartingale. The following Lemma is a minor improvement of Lemma 5 in [48].

Lemma 5.8. Let A = (xn) and B = (yn) be two bounded submartingales.

(i) For a fixed n, the sequence (En(xm\/ym)) _ is increasing, norm bounded

by \\A\\ + \\B\\, and bounded below by xn V yn.

(ii) If, in addition, this sequence converges weakly to some (

then Z = (zn) is a martingale, and it is the least martingale satisfying A^ Z and B ^ Z.

Proof. Let A = (xn) and B = (yn) be two bounded submartingales and n < rn.

We notice that En(xm V ym) ^ (Enxm) V (Enym) ^ xn V yn. Furthermore,

En(xm+i V ym+i) = EnEm(xm+i V ym+\)

^ En(Emxm+1 V Emym+i) ^ En(xm V ym).

Finally, ||£n(a;m V ym)\\ < ||a;mVj/m|| ^ |||xm| + |ym||| < ||A|| + ||£||.

Suppose that w-limm En(xmVym) = zn for each n, and set Z = (zn). First, observe that Z is a martingale. Indeed, for A; ^ n we have

£fc2:n = Ek(w-\imEn{xm V ym)) m—>oo

= w-lim EkEn{xm V ym) = w-lim Ek(xm V ym) = zk. m—»oo m—>oo

Since £„(xm V ym) ^ xnVyn whenever m ^ n, we have f(En(xm V ym)) ^

f(xnVyn) for each positive functional / in X*. So, we have f(zn) ^ f{xnVyn) for each positive functional / in X*. Thus, we have zn ^ xn V y„ for all n.

Thus, Z ^ A and Z ^ B. On the other hand, suppose that Z = (zn) is a

71 martingale such that Z ^ A and Z ^ B. Then zm ^ xm V ym for all m, so that zn = Enzm ^ En(xm V ym) for all m ^ n. As w-limm -E„(xm V ym) = zn, this yields f(zn) ^ /(-£„) for every positive functional / in X*. So, we have so zn ^ zn, that Z ^ Z. Finally, observe that

||z„|| ^ liminf £n(xm Vym) < ||X|| + ||Y||

m—>oo so that Z is bounded. •

Corollary 5.9. Let A = (xn) be a bounded submartingale.

(i) For a fixed n, the sequence (En(\xm\)) _ is increasing, norm bounded

by 2\\A\\, and bounded below by \xn\.

(ii) If, in addition, this sequence converges weakly to some (zn) for each n,

then Z = (zn) is a martingale, and it is the least martingale satisfying A ^ Z and -A ^ Z.

Lemma 5.10. Let A = (xn) £ M such that the limit w-linim-En|xm| exists for each n, denote it zn. Put Z = (zn), then Z is a martingale, Z = \A\, and \\Z\\ = \\A\\.

Proof By Corollary 5.9, Z is indeed a bounded positive martingale and Z =

\A\. From Remark 5.3, we have ||.4|| = supm||xm|| = lim ||xm||. m—>oo

|zn|| = sup f(zn) = sup lim f(En\xm\) ll/ll=i,/>o ||/||=i,/>om->0°

^ lim £„|xm| < lim ||xm|| = ||J4||. m—>oo ro—>oo

On the other hand, for n ^ m we have \xn\ = |-En:rm| < En\xm\. It follows

72 that

||zn|| = sup f(zn) = sup lim f(En\xm\) ll/ll=i,/^o 11/11=1,/^)™-*°°

^ sup lim /(|x„|) = sup f(\xn\) = \\xn\\. ||/||=i,/^o™-°° ll/||=i,/^o

This yields \\Z\\ = \\A\\. D

Theorem 5.11 (Troitsky [48]). If X is a KB-space then M(X,(En)) is a Banach lattice with lattice operations given by

(Ay B)n= lim En{xmVym) m—>oo

(A A B)n = lim En(xm A ym) m—>oo

(A+)n = lim £n(x+) (5.1)

(A~)n = lim £„(x~) m—>oo

|>l|n = lim 2?n|a;m| m—>oo w/iere A = (xn) and B = (yn) are in M(X, (En)).

Proof. Let A = (xn) and B = (yn) be two bounded martingales in M(X, (En)).

It follows from Lemma 5.8(i) that for every n the sequence (En(xm Vym)) _ is increasing in m and norm bounded, hence it converges. It follows then by

Lemma 5.8(h) that A\/B exists and is given by (AVB)n = lim En(xmVym). m—>oo

The other formulae in (5.1) follow immediately. This proves that M{X, (En)) is a lattice. Finally, |||-<4||| = ||v4|| by Lemma 5.10, so that M(X,(En)) is a Banach lattice. • Theorem 5.11 tells us that when X is a KB-spa.ce, the bounded martin­ gale space is a Banach lattice. So, since LP(P) is a KB-space, the classical 7 bounded martingale space Mp = Mp(fi, J , (J-n)-,P) is a Banach lattice. We now characterize those Banach spaces with a filtration for which the bounded martingale space is a Banach lattice. We use the following well known fact.

73 Lemma 5.12. Let X be a Banach lattice and (xn) be an increasing or de­ creasing sequence in X. Then if xn converges to some x weakly in X, then xn converges to x in norm.

Proof. Without loss of generality, assume xn J, and xn —>• 0. For each positive functional, f(xn) ^ 0. Since X* separates points of X, we have xn ^ 0 for each n. Thus, since weak closure and norm closure of a convex set are the same, for each e > 0, there exists a convex combination \ix± + • • • + Xnxn such that ||AiXi H h Anzn|| ^ e. By monotonicity of norm, \\xn\\ ^ ||AiXi H h

An^n|| ^ £• This proves xn —> 0. •

Theorem 5.13. If X is an order continuous Banach lattice and (En) is a filtration on X, then the following statements are equivalent.

(i) M(X,(En)) is a Banach lattice,

(ii) for each n, (En\xm\)m converges weakly for each (xn) £ M(X, (En)).

(Hi) for each n, (En\xm\)m converges in norm for each (xn) e M(X, (En)).

Proof, (i) =>• (ii) Suppose M(X, (En)) is a Banach lattice and let A = (xn) G

M(X,(En)). Thus, \A\ exists. Put \A\ = (yn) G M(X,(En)). This implies , that, for each m ^ n, we have 0 ^ £ n|arm| ^ Enym = yn since \xm\ ^ ym for each m and (En) is a filtration. Since X is order continuous, every or­ der interval is weakly compact. So, for each fixed n, there is a subsequence

(En\xmi\)mi^n that converges weakly to some vector zn. It follows from Corol­ lary 5.9(i) that (En\xm\)m^n is increasing. So, for each positive functional /, we have f(En\xm\) —> f(zn) as m —> oo. It is well known that for each func­ tional /, the positive part /+ and negative part /~ exist and f = f+ — f~ • Thus, for each functional /, we have

+ + f(En\xm\) = f (En\xm\) - f-(En\xm\) - f (zn) - f-(zn) = f(zn)

as m —> co. Hence, for each n, (En\xm\)m^n converges weakly.

(ii) =>• (Hi) Suppose for each n, (En\xm\)m^n converges weakly. By Corol­ lary 5.9(i), for each n, (En\xm\)m^n is increasing. So, by Lemma 5.12, for each

74 n, (En\xm\)m^n converges in norm. (Hi) => (ii) => (i) is straightforward from Corollary 5.9. •

We will use the following standard fact.

Proposition 5.14. Every band projection on a vector lattice is a lattice ho- momorphism.

Proof. Let P : X —» X be a band projection and let x E X. Then x = y + z for some y € kerP and z € RangeP with |x| A \y\ = 0. Thus, P\x\ = P\y\ + P\z\ = P\z\ = \z\ and \Px\ = \Py + Pz\ = \Pz\ = \z\. Hence, P\x\ = \Px\. D

Theorem 5.15. If X is a Banach lattice and (En) is a filtration on X such s that En is a band projection for each n, then M(X, (£«)) i> & Banach lattice with coordinate wise lattice operations.

Proof. Let A = (xn) be a bounded martingale. Then, by Proposition 5.14, x r a -Enl^ml = \Enxm\ = \ n\ f° U Tu ^ n. Thus, (\xn\) is a bounded martingale. It is clearly the smallest bounded martingale among the upper bounds of A.

So, \A\ = (|xn|). So, M(X, (En)) is a Banach lattice with coordinate wise lattice operations. •

5.4 Counterexamples for non-Banach lattice martingale spaces

V. Troitsky asked in [48] whether the space of all bounded martingales M (X, (En)) is a Banach lattice whenever the underlying space is a Banach lattice. This question has a negative answer even if the underlying space is a Banach lat­ tice with order continuous norm. In this section, we present examples where

M(X, (En)) is not a Banach lattice. Namely, we find an order continuous Ba­ nach lattice X, a filtration (En), and a bounded martingale A G M(X, (En)) such that A has no modulus. It implies that A is not regular and that M is not a Banach lattice.

75 Example 5.16. Our construction is based on ideas from [20]. Suppose that

Y is a Banach space, and let a sequence (an)^=0 in Y be an infinite tree, that is, a sequence with the property an = \{a,2n+i + a2n+2) for every n ^ 0. Now define a sequence (xn) in Za([0,1], Y) via

x0 = aoX[o,i) xi = aiX[o,i) + a2X[i,i)

x2 = a3X[o,i) + a4X[i|)+a5X[|,|) + a6X[|,i)

2fc+l_!

i=2fc

1 k fc+1 where IKl.x = [i^-, ^f^ ) for i = 2 , 2* + 1,.., 2 - 1. Then (xn) is a martingale in the sense of [20] relative to the dyadic filtration of [0,1]. Now, suppose that Y is a Banach lattice. Then Li([0, l],Y) also is a Banach lattice. Moreover, if Y is order continuous, then so is Li([0,1], Y). Next, we define a sequence of projections on Lj([0, l],Y) as follows. For /GL!([0,1],Y) we put

Eof = (/o/)X[o,i Eif (^/)Xto,i) + (/i/)X[i,:

= 4 E2f (Jo* /)X[o.i) + (if /)X[i,l) + (if /)X[f ,f) + ill /)%!,

2fc+i_1 £*/ 2/ /x Ik,i-l

76 where /^ = [i=2^i, i=J^±l) for i = 2fc,2fc + l,...^1 - 1. It is easy to see that (En) is a filtration on Lx([0, l],y), and (xn) is a martingale in L\ ([0,1], y) relative to this filtration.

Now put y = Co and let X = L\ ([0,1], c0). Since CQ is an order continuous Banach lattice, so is X. Let (xk) be defined as above with

ao = (0,...)

fll = (l,0,...) a2 = (-l,0,...) a3 = (l,l,0,...) a4 = (l,-l,0...) a5 = (-1,1,0,...) a6 = (-1,-1,0,...)

In other words, xn = (r1;..., rn,0,...), where Tk is the fc-th Rademacher function. Then ||xn|| = 1 for every n. However,

Eo\xn\ = {l,...,l,0...) n times

It follows from Theorem 5.13 that (xn) is a martingale with no modulus.

Example 5.17. In this example we construct a filtration (En) on CQ such that M(co, (En)) is not a Banach lattice.

As usually, an operator T 6 -L(c0) can be represented by an infinite matrix where the j-column is Tej. For n = 0,1, 2,..., put

"l 0

1 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 0

77 with 2n l's in the upper-left corner. In other words,

Enei = et if i ^ 2n, and Ene2k = Ene2k-i = \{e2k + e2k-i) when n < k.

Note that EQ has no l's at all. It is easy to see that (En) is a dense filtration. Furthermore, for n = 0,1, 2,..., put

xn = (-1,1,..., -1,1, 0,...). s v

It is easy to see that (xn) is a bounded martingale relative to (En). On the other hand,

E0\xn\ = (1,1,..., 1,1,0,...). 2n

Hence, Eo\xn\, so that (xn) has no modulus by Theorem 5.13. Therefore,

M(c0, (En)) is not a Banach lattice.

5.5 How does X sit in M(X, {En))l

Throughout this section we assume that X is a Banach lattice, (En) is a dense filtration on X (i.e, Enx —• x for each x € X), and M(X, (En)) is the space of bounded martingales. It was observed in Section 8 of [48] that in this case a martingale (xn) converges if and only if it is fixed, i.e., there exists x £ X such that xn = Enx for every n; also xn —> x. Thus, if for every x € X we define a map (p as

Hence, we can identify X with a closed subspace

Lemma 5.18. For all x,y £ X, we have x ^ y if and only if ip(x) ^

Proof. It is clear that x ^ y implies (p(x) ^ x and Eny —> y. Take any x,y £ X and put xn = £ni

78 and yn = Eny for all n. Since the lattice operations are continuous, we have xn V yn -^ xV y. Hence, for every n, we have

lim En(xm V ym) = En( lim xm V ym) = En(x V y). m—>oo m—>oo

It follows from Lemma 5.8 that

ip(x) V ip(y) = {Enx)n V {Eny)n = (xn)n V (j/n)n

= ( lim En(xm V ym)) = (En( lim xm V ym)) ra—>oo m—nx)

= (En{xVy)) = ^(xVy)

So, we have (p(x) V ip(y) = y?(x V y). To prove (p(x Aj/) = <^(x) A ip(y), we use Lemma 2.3. Indeed, since (x) A (y) = —((—x) V (—?/)) from Lemma 2.3, we have

D

Recall that a subset D in a Banach lattice X is said to be order bounded if D C [—x,x] for some x e A+, and almost order bounded if, for every

£ > 0, there exists x e X+ such that Z) C [—x,x] + eB(0,1), where B(0,1) stands for the unit ball of X. One can easily see that a subset of Iq(-P) is uniformly integrable if and only if it is almost order bounded. If X is a Banach lattice with an order continuous norm then it follows from Theorems 2.6 and Theorem 2.7 that every almost order bounded set in X is relatively weakly compact. This provides us a generalization of Doob Convergence Theorem.

Corollary 5.19 ([48]). Every almost order bounded martingale in a Banach lattice with order continuous norm is fixed.

We will use the preceding Corollary in the proof of the following lemma.

79 Lemma 5.20. If X is order continuous and M(X, (En)) is a Banach lattice, then M(X, (En)) is order complete and

Proof. First, show that M(X,(En)) is order complete. Suppose that 0 ^ [ ] A^ K A in M(X, (£•„)). Put A = (xn) and A™ = (x^). Then 0 < x n K xn for every n. Since X is order continuous, for every n, there exists yn such that we have i„ —> yn. If B = (yn), then B is a martingale. Indeed,

Enym = En{\imx^) = UmEnx£> = lunx^ = yn. Oi OL CX.

It follows from 0 < B < A that B is bounded, hence B E M(X, (En)). Put

M0 = (p(X). By Lemma 5.18, M0 is lattice homeomorphic to X.

We show that M0 is an ideal in M(X, (En)). Suppose that 0 ^ A ^ B for some A E M(X, (En)) and B E M0. Put 4 = (xn) and 5 = (yn). Then

0 ^ x„ ^ yn for every n and there exists y E X such that y„ = Eny for all n.

Fix e > 0. It follows from t/n —> y that there exists n0 such that \\yn — y\\ < £ whenever n ^ n0. It follows from Theorem 2.3 (vi) that

\xn -xnAy\ = \xn Ayn-xnAy\^ \yn - y\,

so that \\xn — xnAy\\ < e. It follows from xnAy E [0, y] that xn E [0, y]+eB(0,1) for all n ^ n0. Therefore, (xn) is almost order bounded. Hence, it converges by Corollary 5.19. Hence, A E M0. Notice that if B E M0, then \B\ E M0.

Indeed, suppose B E M0. Then B = (Enx)n for some x E X and Enx —> x as n —*• oo. This implies |-Enx| —> \x\ as n —> oo since ||x| — \y\\ ^ |x + y\ by Theorem 2.3 (v) and the norm is monotone. Thus, for each m, Em\Enx\ converges to Em\x\. So, by Corollary 5.9, |5| = (En\x\)n. Therefore, \B\ E M0.

Now suppose that B E M0 and A E M(X, (En)) such that \A\ ^ \B\. Then + + 0 ^ A ,A~ < \B\, so that A ,A~ E M0 and, therefore, A E M0. Thus, M0 is an ideal. •

Theorem 5.21. // X is a KB-space, then

M(X,(En)).

80 Proof. By Theorems 5.11 and 5.20, M(X, (£„)) is an order complete Banach lattice. Again, denote M0 = tp(X). By Lemma 5.20, (p(X) is an ideal in

M{X, {En)). It is left to show that M0 is a band because every band in an order continuous lattice is a projection band by Theorem 2.4.

To show that M0 is a band, suppose that 0 ^ A^ f A for some net

(A^) in M0 and some A E M(X,(En)). Put A = (xn) and A^ = «). Let A^ = (p(x^) for some x^ E X. Since ip is an isometry, we have ||x^|| = ||J4^|| ^ ||-A|| for every a, hence the net (x^) is norm bounded. Since X is a KB-spa.ce, by definition, this net converges in norm to some (a) y E X. It follows also that x | V in X. Put 5 = y implies tp(x^) —> y(y). This implies r limaXn —> yn f° every n. Together with i„ ^ xn, this implies yn ^ xn, so that J5 < A. Thus, 4 = 5, so that A E M0. D

It was proved in [48] that if a martingale A = (xn) in M(X, (£„)) converges weakly, weakly as n —> oo, then A = (Enx). It is easy to see that when the underlying Banach lattice X is reflexive, every bounded martingale converges weakly. It is also well-known that a reflexive Banach lattice is a KB-space. Thus, we have the following simple consequence of Theorem 5.21.

Corollary 5.22. If X is a reflexive Banach lattice, then every bounded mar­ tingale is fixed. In particular, if X is a reflexive Banach lattice and (En) is dense then M(X,(En)) = X.

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