DOCSLIB.ORG

Lectures on Functional Calculus

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lectures on Functional Calculus Markus Haase Lectures on Functional Calculus { 21st International Internet Seminar { November 29, 2017 CAU Press 2017 Contents 1 Holomorphic Functions of Bounded Operators ............ 7 1.1 Polynomial Functional Calculus . .7 1.2 Power Series Functional Calculi . .8 1.3 The Dunford{Riesz Calculus . 10 1.4 From Bounded to Unbounded Operators . 18 1.5 Supplement: Functional Calculus for Matrices . 19 Exercises . 22 2 Unbounded Operators and Abstract Functional Calculus . 25 2.1 Unbounded Operators. 25 2.2 Multiplication Operators . 27 2.3 Functional Calculus for Multiplication Operators . 29 2.4 Abstract Functional Calculus (I) | Definition . 30 2.5 Abstract Functional Calculus (II) | Generators . 34 Exercises . 36 3 Borel Functional Calculus ................................ 39 3.1 Normal Operators on Hilbert Spaces . 39 3.2 Measurable Functional Calculus . 41 3.3 Projection-Valued Measures . 43 3.4 Borel Functional Calculus . 46 3.5 Supplement: Commutation Results . 49 Exercises . 51 4 The Spectral Theorem .................................... 55 4.1 Three Versions of the Spectral Theorem . 55 4.2 Proof: Bounded Self-Adjoint Operators . 58 4.3 Proof: From Continuous Functions to Multiplication Operators 59 4.4 Proof: From Bounded to Unbounded Operators . 62 4.5 Supplement: Normal Operators . 64 Exercises . 66 v vi Contents 5 Fourier Analysis .......................................... 69 5.1 Strongly Continuous Semigroup Representations . 69 5.2 The Regular Representations. 71 5.3 Averaging a Representation and the Convolution of Measures 73 5.4 The Fourier Transform . 77 Exercises . 81 6 Integral Transform Functional Calculi .................... 85 6.1 The Fourier{Stieltjes Calculus . 85 6.2 Bounded C0-Semigroups and the Hille{Phillips Calculus . 88 6.3 General C0-Semigroups and C0-Groups . 92 6.4 Supplement: Continuity Properties and Uniqueness . 96 6.5 The Heat Semigroup on Rd .............................. 99 6.6 Supplement: Subordinate Semigroups . 103 Exercises . 106 Appendix .....................................................1001 A.1 Notational and Terminological Conventions . 1001 A.2 Calculus in Banach Spaces . 1005 A.3 Vector-valued Holomorphic Functions . 1010 A.4 Unbounded Operators. 1014 A.5 Operators on Hilbert Space . 1024 A.6 The Bochner Integral . 1029 A.7 Complex Measures. 1034 Chapter 1 Holomorphic Functions of Bounded Operators 1.1 Polynomial Functional Calculus Let X be a Banach space (over C) and A 2 L(X) a bounded operator on it. Then for every polynomial 2 n p = a0 + a1z + a2z + ··· + anz 2 C[z] (1:1) we can form the (likewise bounded) operator 2 n p(A) := a0 + a1A + a2A + ··· + anA 2 L(X): The space of polynomials C[z] is a unital algebra and the mapping Ψ : C[z] !L(X); p 7! Ψ(p) := p(A) is a (unital) algebra homomorphism. In other words, Ψ is a representation of the unital algebra C[z] on the vector space X by bounded operators. Obviously, any representation Ψ of C[z] on X by bounded operators is of the form above: simply define A := Ψ(z) and find Ψ(p) = p(A) for all p 2 C[z]. One may call Ψ a calculus since it reduces calculations with operators to calculations with other (here: formal) objects. Strictly speaking, however, this calculus is not (yet) a functional calculus, since a polynomial is not a function. It can become a function when it is interpreted as one, but this needs the specification of a domain, say D ⊆ C. Let us denote, for a polynomial p 2 C[z], its interpretation as a function on D by pjD. Then the indeterminate, which we have called \z" here, is mapped to the function zjD : D ! C; z 7! z: Here and in all what follows, we shall denote this function by z. If p has the form given in (1.1) then 7 8 1 Holomorphic Functions of Bounded Operators 2 n pjD = a0 + a1z + a2z + ··· + anz : Let us take D = C, which is somehow the most natural choice. Then the mapping p 7! pjC is injective. Hence we can take its inverse and compose it with Ψ from above to obtain a unital algebra homomorphism Φ : fpolynomial functions on Cg ! L(X);Φ(pjC) = Ψ(p): Now this is indeed a functional calculus, called the polynomial functional calculus. Note that A = Φ(z) and Φ is uniquely determined by this value. 1.2 Power Series Functional Calculi So far, the property that A is bounded was not used at all. Everything was pure algebra. Taking boundedness into account we can extend the polynomial functional calculus to a functional calculus of entire functions. Namely, each entire function f 2 Hol(C) has a unique power series representation 1 2 n X n f = a0 + a1z + a2z + ··· + anz ··· = anz : n=0 The convergence is absolute and locally uniform, so that in particular 1 X n janj r < 1 for each r 2 R+. n=0 Hence, we can define 1 X n f(A) := Φ(f) := anA 2 L(X); n=0 the series being absolutely convergent in the Banach space L(X). Clearly, if f is a polynomial (function), this definition of f(A) agrees with the previous. The so obtained mapping Φ : Hol(C) !L(X); f 7! f(A) is a homomorphism of unital algebras, because the usual rules of manipulation with absolutely convergent series are the same in C and in L(X) (actually in all Banach algebras). Notably, the multiplicativity of Φ follows from the fact that the Cauchy product formula 1.2 Power Series Functional Calculi 9 1 1 1 n X X XX an bn = ajbn−j n=0 n=0 n=0 j=0 for absolutely convergent series holds in C as well as in L(X). The functional calculus for entire functions works for every bounded operator A. If we take kAk or even all the norms kAnk, n 2 N, into account, we can extend the calculus further, for instance as follows: n −n Fix r > 0 such that M(A; r) := supn≥0 kA kr < 1. (One can take r = kAk, but in general r < kAk is possible.) Abbreviate Dr := B(0; r) (and 1 D = D1), and denote by A+(Dr) the set of functions f : Dr ! C that can be represented as a power series 1 X n f(z) = anz (jzj ≤ r) (1:2) n=0 such that 1 X n kfk 1 := kfk 1 := janj r < 1: (1:3) A+ A+(Dr ) n=0 1 Note that each f 2 A+(Dr) is holomorphic on Dr and continuous on Dr = B[0; r]. In particular, the coefficients an are uniquely determined by the function f, so kfk 1 is well defined. Using the Cauchy product formula, A+ 1 it is easy to see that A+(Dr) is a unital Banach algebra. (Exercise 1.2). 1 For f 2 A+(Dr) with representation (1.2) we can now define 1 X n f(A) := Φ(f) := anA : n=0 The condition (1.3) ensures that this power series is absolutely convergent and computations as before show that 1 Φ :A+(Dr) !L(X); f 7! f(A) is a homomorphism of unital algebras. Moreover, one has the norm estimate 1 kf(A)k ≤ M(A; r) kfk 1 (f 2 A ( r)) A+ + D as is easily seen. Remark 1.1. The case r = 1 is particularly important. An operator with M(A; 1) < 1 is called power bounded. So, power bounded operators have 1 a (bounded) A+(D)-calculus, a fact that will become important later in this course. 10 1 Holomorphic Functions of Bounded Operators Note that in the above construction one has A = Φ(z), and by this condi- tion Φ is uniquely determined as a continuous unital algebra homomorphism 1 A+(Dr) !L(X). 1.3 The Dunford{Riesz Calculus The above power series functional calculi for an operator A use only informa- tion about the asymptotic behaviour of the norms kAnk. In this section we shall construct a holomorphic functional calculus for A that is based purely on the location of its spectrum. Review of elementary spectral theory Recall from an elementary functional analysis course that the spectrum of A 2 L(X) is the set s(A) := fλ 2 C : λ − A is not invertibleg: (We write λ − A instead of λI − A.) The complement r(A) := C n s(A), an open subset of C, is called the resolvent set of A. The mapping R(·;A): r(A) !L(X);R(λ, A) := (λ − A)−1 is the resolvent. It satisfies the resolvent identity R(λ, A) − R(µ, A) = (µ − λ)R(λ, A)R(µ, A)(λ, µ 2 r(A)) (1:4) and is holomorphic. (See Appendix A.3 for an introduction to vector-valued holomorphic mappings.) P1 −(n+1) n A number λ 6= 0 belongs to r(A) whenever the series n=0 λ A converges, in which case this series equals R(λ, A). (This follows from the Neumann series theorem.) In particular, λ 2 r(A) if jλj > kAk and then 1 kR(λ, A)k ≤ : jλj − kAk Hence, the spectrum is compact and not empty (by Liouville's theorem), except in the case X = f0g. The spectral radius r(A) := supfjλj : λ 2 s(A)g satisfies r(A) ≤ kAk. 1.3 The Dunford{Riesz Calculus 11 Dunford{Riesz{Calculus, \baby" version We start with a \baby" version of the Dunford{Riesz calculus and take s > r(A), i.e., s(A) ⊆ Ds. For a function f 2 Hol(Ds) define 1 Z Φ(f) := f(z)R(z; A) dz (1:5) 2pi γr where 0 < r < s is such that still s(A) ⊆ Dr and γr is the contour which traverses the boundary of Dr once in counterclockwise direction.
Load more

Recommended publications
  • An Elementary Proof of the Spectral Radius Formula for Matrices
    AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES JOEL A. TROPP Abstract. We present an elementary proof that the spectral ra- dius of a matrix A may be obtained using the formula 1 ρ(A) = lim kAnk =n; n!1 where k · k represents any matrix norm. 1. Introduction It is a well-known fact from the theory of Banach algebras that the spectral radius of any element A is given by the formula ρ(A) = lim kAnk1=n: (1.1) n!1 For a matrix, the spectrum is just the collection of eigenvalues, so this formula yields a technique for estimating for the top eigenvalue. The proof of Equation 1.1 is beautiful but advanced. See, for exam- ple, Rudin's treatment in his Functional Analysis. It turns out that elementary techniques suffice to develop the formula for matrices. 2. Preliminaries For completeness, we shall briefly introduce the major concepts re- quired in the proof. It is expected that the reader is already familiar with these ideas. 2.1. Norms. A norm is a mapping k · k from a vector space X into the nonnegative real numbers R+ which has three properties: (1) kxk = 0 if and only if x = 0; (2) kαxk = jαj kxk for any scalar α and vector x; and (3) kx + yk ≤ kxk + kyk for any vectors x and y. The most fundamental example of a norm is the Euclidean norm k·k2 which corresponds to the standard topology on Rn. It is defined by 2 2 2 kxk2 = k(x1; x2; : : : ; xn)k2 = x1 + x2 + · · · + xn: Date: 30 November 2001.
    [Show full text]
  • Stat 309: Mathematical Computations I Fall 2018 Lecture 4
    STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 4 1. spectral radius • matrix 2-norm is also known as the spectral norm • name is connected to the fact that the norm is given by the square root of the largest eigenvalue of ATA, i.e., largest singular value of A (more on this later) n×n • in general, the spectral radius ρ(A) of a matrix A 2 C is defined in terms of its largest eigenvalue ρ(A) = maxfjλij : Axi = λixi; xi 6= 0g n×n • note that the spectral radius does not define a norm on C • for example the non-zero matrix 0 1 J = 0 0 has ρ(J) = 0 since both its eigenvalues are 0 • there are some relationships between the norm of a matrix and its spectral radius • the easiest one is that ρ(A) ≤ kAk n for any matrix norm that satisfies the inequality kAxk ≤ kAkkxk for all x 2 C , i.e., consistent norm { here's a proof: Axi = λixi taking norms, kAxik = kλixik = jλijkxik therefore kAxik jλij = ≤ kAk kxik since this holds for any eigenvalue of A, it follows that maxjλij = ρ(A) ≤ kAk i { in particular this is true for any operator norm { this is in general not true for norms that do not satisfy the consistency inequality kAxk ≤ kAkkxk (thanks to Likai Chen for pointing out); for example the matrix " p1 p1 # A = 2 2 p1 − p1 2 2 p is orthogonal and therefore ρ(A) = 1 but kAkH;1 = 1= 2 and so ρ(A) > kAkH;1 { exercise: show that any eigenvalue of a unitary or an orthogonal matrix must have absolute value 1 Date: October 10, 2018, version 1.0.
    [Show full text]
  • Inverse and Implicit Function Theorems for Noncommutative
    Inverse and Implicit Function Theorems for Noncommutative Functions on Operator Domains Mark E. Mancuso Abstract Classically, a noncommutative function is defined on a graded domain of tuples of square matrices. In this note, we introduce a notion of a noncommutative function defined on a domain Ω ⊂ B(H)d, where H is an infinite dimensional Hilbert space. Inverse and implicit function theorems in this setting are established. When these operatorial noncommutative functions are suitably continuous in the strong operator topology, a noncommutative dilation-theoretic construction is used to show that the assumptions on their derivatives may be relaxed from boundedness below to injectivity. Keywords: Noncommutive functions, operator noncommutative functions, free anal- ysis, inverse and implicit function theorems, strong operator topology, dilation theory. MSC (2010): Primary 46L52; Secondary 47A56, 47J07. INTRODUCTION Polynomials in d noncommuting indeterminates can naturally be evaluated on d-tuples of square matrices of any size. The resulting function is graded (tuples of n × n ma- trices are mapped to n × n matrices) and preserves direct sums and similarities. Along with polynomials, noncommutative rational functions and power series, the convergence of which has been studied for example in [9], [14], [15], serve as prototypical examples of a more general class of functions called noncommutative functions. The theory of non- commutative functions finds its origin in the 1973 work of J. L. Taylor [17], who studied arXiv:1804.01040v2 [math.FA] 7 Aug 2019 the functional calculus of noncommuting operators. Roughly speaking, noncommutative functions are to polynomials in noncommuting variables as holomorphic functions from complex analysis are to polynomials in commuting variables.
    [Show full text]
  • Hilbert's Projective Metric in Quantum Information Theory D. Reeb, M. J
    Hilbert’s projective metric in quantum information theory D. Reeb, M. J. Kastoryano and M. M. Wolf REPORT No. 31, 2010/2011, fall ISSN 1103-467X ISRN IML-R- -31-10/11- -SE+fall HILBERT’S PROJECTIVE METRIC IN QUANTUM INFORMATION THEORY David Reeb,∗ Michael J. Kastoryano,† and Michael M. Wolf‡ Department of Mathematics, Technische Universit¨at M¨unchen, 85748 Garching, Germany Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark (Dated: August 15, 2011) We introduce and apply Hilbert’s projective metric in the context of quantum information theory. The metric is induced by convex cones such as the sets of positive, separable or PPT operators. It provides bounds on measures for statistical distinguishability of quantum states and on the decrease of entanglement under LOCC protocols or other cone-preserving operations. The results are formulated in terms of general cones and base norms and lead to contractivity bounds for quantum channels, for instance improving Ruskai’s trace-norm contraction inequality. A new duality between distinguishability measures and base norms is provided. For two given pairs of quantum states we show that the contraction of Hilbert’s projective metric is necessary and sufficient for the existence of a probabilistic quantum operation that maps one pair onto the other. Inequalities between Hilbert’s projective metric and the Chernoff bound, the fidelity and various norms are proven. Contents I. Introduction 2 II. Basic concepts 3 III. Base norms and negativities 7 IV. Contractivity properties of positive maps 9 V. Distinguishability measures 14 VI. Fidelity and Chernoff bound inequalities 21 VII. Operational interpretation 24 VIII.
    [Show full text]
  • The Notion of Observable and the Moment Problem for ∗-Algebras and Their GNS Representations
    The notion of observable and the moment problem for ∗-algebras and their GNS representations Nicol`oDragoa, Valter Morettib Department of Mathematics, University of Trento, and INFN-TIFPA via Sommarive 14, I-38123 Povo (Trento), Italy. [email protected], [email protected] February, 17, 2020 Abstract We address some usually overlooked issues concerning the use of ∗-algebras in quantum theory and their physical interpretation. If A is a ∗-algebra describing a quantum system and ! : A ! C a state, we focus in particular on the interpretation of !(a) as expectation value for an algebraic observable a = a∗ 2 A, studying the problem of finding a probability n measure reproducing the moments f!(a )gn2N. This problem enjoys a close relation with the selfadjointness of the (in general only symmetric) operator π!(a) in the GNS representation of ! and thus it has important consequences for the interpretation of a as an observable. We n provide physical examples (also from QFT) where the moment problem for f!(a )gn2N does not admit a unique solution. To reduce this ambiguity, we consider the moment problem n ∗ (a) for the sequences f!b(a )gn2N, being b 2 A and !b(·) := !(b · b). Letting µ!b be a n solution of the moment problem for the sequence f!b(a )gn2N, we introduce a consistency (a) relation on the family fµ!b gb2A. We prove a 1-1 correspondence between consistent families (a) fµ!b gb2A and positive operator-valued measures (POVM) associated with the symmetric (a) operator π!(a). In particular there exists a unique consistent family of fµ!b gb2A if and only if π!(a) is maximally symmetric.
    [Show full text]
  • Algebra Homomorphisms and the Functional Calculus
    Pacific Journal of Mathematics ALGEBRA HOMOMORPHISMS AND THE FUNCTIONAL CALCULUS MARK PHILLIP THOMAS Vol. 79, No. 1 May 1978 PACIFIC JOURNAL OF MATHEMATICS Vol. 79, No. 1, 1978 ALGEBRA HOMOMORPHISMS AND THE FUNCTIONAL CALCULUS MARC THOMAS Let b be a fixed element of a commutative Banach algebra with unit. Suppose σ(b) has at most countably many connected components. We give necessary and sufficient conditions for b to possess a discontinuous functional calculus. Throughout, let B be a commutative Banach algebra with unit 1 and let rad (B) denote the radical of B. Let b be a fixed element of B. Let έ? denote the LF space of germs of functions analytic in a neighborhood of σ(6). By a functional calculus for b we mean an algebra homomorphism θr from έ? to B such that θ\z) = b and θ\l) = 1. We do not require θr to be continuous. It is well-known that if θ' is continuous, then it is equal to θ, the usual functional calculus obtained by integration around contours i.e., θ{f) = -±τ \ f(t)(f - ]dt for f eέ?, Γ a contour about σ(b) [1, 1.4.8, Theorem 3]. In this paper we investigate the conditions under which a functional calculus & is necessarily continuous, i.e., when θ is the unique functional calculus. In the first section we work with sufficient conditions. If S is any closed subspace of B such that bS Q S, we let D(b, S) denote the largest algebraic subspace of S satisfying (6 — X)D(b, S) = D(b, S)f all λeC.
    [Show full text]
  • The Weyl Functional Calculus F(4)
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF FUNCTIONAL ANALYSIS 4, 240-267 (1969) The Weyl Functional Calculus ROBERT F. V. ANDERSON Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Communicated by Edward Nelson Received July, 1968 I. INTRODUCTION In this paper a functional calculus for an n-tuple of noncommuting self-adjoint operators on a Banach space will be proposed and examined. The von Neumann spectral theorem for self-adjoint operators on a Hilbert space leads to a functional calculus which assigns to a self- adjoint operator A and a real Borel-measurable function f of a real variable, another self-adjoint operatorf(A). This calculus generalizes in a natural way to an n-tuple of com- muting self-adjoint operators A = (A, ,..., A,), because their spectral families commute. In particular, if v is an eigenvector of A, ,..., A, with eigenvalues A, ,..., A, , respectively, and f a continuous function of n real variables, f(A 1 ,..., A,) u = f(b , . U 0. This principle fails when the operators don’t commute; instead we have the Uncertainty Principle, and so on. However, the Fourier inversion formula can also be used to define the von Neumann functional cakuius for a single operator: f(4) = (37F2 1 (Sf)(&) exp( -2 f A,) 4 E’ where This definition generalizes naturally to an n-tuple of operators, commutative or not, as follows: in the n-dimensional Fourier inversion 240 THE WEYL FUNCTIONAL CALCULUS 241 formula, the free variables x 1 ,..,, x, are replaced by self-adjoint operators A, ,..., A, .
    [Show full text]
  • Banach Algebras
    Banach Algebras Yurii Khomskii Bachelor Thesis Department of Mathematics, Leiden University Supervisor: Dr. Marcel de Jeu April 18, 2005 i Contents Foreword iv 1. Algebraic Concepts 1 1.1. Preliminaries . 1 1.2. Regular Ideals . 3 1.3. Adjoining an Identity . 4 1.4. Quasi-inverses . 8 2. Banach Algebras 10 2.1. Preliminaries of Normed and Banach Algebras . 10 2.2. Inversion and Quasi-inversion in Banach Algebras . 14 3. Spectra 18 3.1. Preliminaries . 18 3.2. Polynomial Spectral Mapping Theorem and the Spectral Radius Formula . 22 4. Gelfand Representation Theory 25 4.1. Multiplicative Linear Functionals and the Maximal Ideal Space . 25 4.2. The Gelfand Topology . 30 4.3. The Gelfand Representation . 31 4.4. The Radical and Semi-simplicity . 33 4.5. Generators of Banach algebras . 34 5. Examples of Gelfand Representations 36 5.1. C (X ) for X compact and Hausdorff . 36 5.2. C 0(X ) for X locally compact and Hausdorff. 41 5.3. Stone-Cecˇ h compactification . 42 5.4. A(D) . 44 5.5. AC (Γ) . 46 5.6. H 1 . 47 ii iii Foreword The study of Banach algebras began in the twentieth century and originated from the observation that some Banach spaces show interesting properties when they can be supplied with an extra multiplication operation. A standard exam- ple was the space of bounded linear operators on a Banach space, but another important one was function spaces (of continuous, bounded, vanishing at infin- ity etc. functions as well as functions with absolutely convergent Fourier series). Nowadays Banach algebras is a wide discipline with a variety of specializations and applications.
    [Show full text]
  • Isospectral Graph Reductions, Estimates of Matrices' Spectra, And
    ISOSPECTRAL GRAPH REDUCTIONS, ESTIMATES OF MATRICES’ SPECTRA, AND EVENTUALLY NEGATIVE SCHWARZIAN SYSTEMS A Thesis Presented to The Academic Faculty by Benjamin Zachary Webb In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Mathematics Georgia Institute of Technology May 2011 ISOSPECTRAL GRAPH REDUCTIONS, ESTIMATES OF MATRICES’ SPECTRA, AND EVENTUALLY NEGATIVE SCHWARZIAN SYSTEMS Approved by: Dr. Leonid A. Bunimovich, Advisor Dr. Dana Randall School of Mathematics College of Computing Georgia Institute of Technology Georgia Institute of Technology Dr. Yuri Bakhtin Dr. Howie Weiss School of Mathematics School of Mathematics Georgia Institute of Technology Georgia Institute of Technology Dr. Luca Dieci Date Approved: March 8, 2011 School of Mathematics Georgia Institute of Technology To my wife, Rebekah. iii ACKNOWLEDGEMENTS In acknowledging the people who have advised, mentored, guided, supported, and helped me throughout my time at Georgia Tech I would like to start with my advisor Dr. Leonid Bunimovich, with whom I have enjoyed working and who has had a significant impact on my understanding of mathematics, mentoring, and academics in general. I am also grateful for Yuri Bakhtin, Luca Dieci, Dana Randall, and Howie Weiss for not only serving as part of my dissertation committee but also guiding and supporting my endeavors while at Georgia Tech. I would also like to thank those in the School of Mathematics at Georgia Tech for the general atmosphere of supportiveness. This is especially true of both Klara Grodzinsky and Rena Brakebill who have guided my teaching efforts over the past years. I would also extend this thanks to Sharon McDowell and Genola Turner who along with my advisor have not only been supportive of myself but of my wife and children who have found themselves in a rather unique situation over the past years of my doctoral studies.
    [Show full text]
  • Riesz-Like Bases in Rigged Hilbert Spaces, in Preparation [14] Bonet, J., Fern´Andez, C., Galbis, A
    RIESZ-LIKE BASES IN RIGGED HILBERT SPACES GIORGIA BELLOMONTE AND CAMILLO TRAPANI Abstract. The notions of Bessel sequence, Riesz-Fischer sequence and Riesz basis are generalized to a rigged Hilbert space D[t] ⊂H⊂D×[t×]. A Riesz- like basis, in particular, is obtained by considering a sequence {ξn}⊂D which is mapped by a one-to-one continuous operator T : D[t] → H[k · k] into an orthonormal basis of the central Hilbert space H of the triplet. The operator T is, in general, an unbounded operator in H. If T has a bounded inverse then the rigged Hilbert space is shown to be equivalent to a triplet of Hilbert spaces. 1. Introduction Riesz bases (i.e., sequences of elements ξn of a Hilbert space which are trans- formed into orthonormal bases by some bounded{ } operator withH bounded inverse) often appear as eigenvectors of nonself-adjoint operators. The simplest situation is the following one. Let H be a self-adjoint operator with discrete spectrum defined on a subset D(H) of the Hilbert space . Assume, to be more definite, that each H eigenvalue λn is simple. Then the corresponding eigenvectors en constitute an orthonormal basis of . If X is another operator similar to H,{ i.e.,} there exists a bounded operator T withH bounded inverse T −1 which intertwines X and H, in the sense that T : D(H) D(X) and XT ξ = T Hξ, for every ξ D(H), then, as it is → ∈ easily seen, the vectors ϕn with ϕn = Ten are eigenvectors of X and constitute a Riesz basis for .
    [Show full text]
  • Wavelets Through a Looking Glass: SATISFACTION GUARANTEED! the World of the Spectrum Remember: Your 30-Day Return Privilege by O
    20% Discount Pre-Pub Offer Forthcoming June 2002! List Price: $59.95 (tent.) Special Price: $47.96 + shipping & handling Wavelets through a Please mention reference # Y404 when placing your order! Offer Valid Until 31 August 2002 Looking Glass Table of Contents The World of the Spectrum List of Figures List of Tables O. Bratteli, University of Oslo, Oslo, Norway Preface P. Jorgensen, University of Iowa, Iowa City, IA 1. Introduction Overture: Why wavelets? * Subband filters This book combining wavelets and the world of the spectrum focuses on recent and sieves * Matrix functions and developments in wavelet theory, emphasizing fundamental and relatively multiresolutions * Qubits: The oracle of timeless techniques that have a geometric and spectral-theoretic flavor. The Feynman and the algorithm of Shor * Chaos exposition is clearly motivated and unfolds systematically, aided by numerous and cascade approximation * Spectral bounds for the transfer and subdivision graphics. operators * Connections to group theory * Wavelet packets * The Gabor transform * Key features of the book: Exercises * Terminology 2. Homotopy Theory and Cascades • The important role of the spectrum of a transfer operator is studied The dangers of navigating the landscape of • wavelets * Homotopy classes of wavelets * Excellent graphics show how wavelets depend on the spectra of the transfer Multiresolution analysis and tight frames * operators Generality of multiresolution analysis * • Key topics of wavelet theory are examined: connected components in the Global homotopy and an index theorem * Cascades in L2(R) * An open problem * variety of wavelets, the geometry of winding numbers, the Galerkin projection Exercises method, classical functions of Weierstrass and Hurwitz and their role in describing the eigenvalue-spectrum of the transfer operator, isospectral 3.
    [Show full text]
  • Isospectral Graph Reductions
    Graphs Reductions Eigenvalue Estimation Summary and Implications Isospectral Graph Reductions Leonid Bunimovich Leonid Bunimovich Isospectral Graph Reductions Networks and Graphs Graphs Reductions Definitions Eigenvalue Estimation Graph Reductions Summary and Implications Main Results Outline 1 Graphs Reductions Networks and Graphs Definitions Graph Reductions Main Results 2 Eigenvalue Estimation Gershgorin’s Theorem Brauer’s Theorem Brualdi’s Theorem 3 Summary and Implications References Leonid Bunimovich Isospectral Graph Reductions Networks and Graphs Graphs Reductions Definitions Eigenvalue Estimation Graph Reductions Summary and Implications Main Results Network Structure Typical real networks are defined by some large graph with complicated structure [2,8,11]. E.coli metabolic network Question: To what extent can this structure be simplified/reduced while maintaining some characteristic of the network? Leonid Bunimovich Isospectral Graph Reductions Networks and Graphs Graphs Reductions Definitions Eigenvalue Estimation Graph Reductions Summary and Implications Main Results The collection of graphs G The graph of a network may or may not be directed, weighted, have multiple edges or loops. Each such graph can be considered a weighted, directed graph without multiple edges possibly with loops. 3 2 1 3 2 Let G be the collection of all such graphs. Leonid Bunimovich Isospectral Graph Reductions Networks and Graphs Graphs Reductions Definitions Eigenvalue Estimation Graph Reductions Summary and Implications Main Results The collection of graphs G Definition A graph G ∈ G is triple G = (V , E, ω) where V is its vertices, E its edges, and ω : E → W where W is the set of edge weights. An important characteristic of a network/graph is the spectrum of its weighted adjacency matrix [1,3,10].
    [Show full text]