Aimed at graduate students, this textbook provides an accessible and comprehensive introduction to operator theory. Rather than discuss the subject in the abstract, this textbook covers the subject through twenty examples of a wide variety of operators, discussing the norm, spectrum, commutant, invariant subspaces, and interesting properties of each operator.
The text is supplemented by over 600 end-of-chapter exercises, designed to help the reader master the topics covered in the chapter, as well as providing an opportunity to further explore the vast operator theory literature. Each chapter also contains well-researched historical facts which place each chapter within the broader context of the development of the field as a whole.
Author(s): Stephan Ramon Garcia, Javad Mashreghi, William T. Ross
Series: Oxford Graduate Texts in Mathematics
Edition: 1
Publisher: Oxford University Press
Year: 2023
Language: English
Pages: 528
Tags: Operator Theory
cover
Operator Theory by Example
Copyright
Dedication
Contents
Preface
Notation
A Brief Tour of Operator Theory
Overview
1 Hilbert Spaces
1.1 Euclidean Space
1.2 The Sequence Space l2
1.3 The Lebesgue Space L2[0, 1]
1.4 Abstract Hilbert Spaces
1.5 The Gram–Schmidt Process
1.6 Orthonormal Bases and Total Orthonormal Sets
1.7 Orthogonal Projections
1.8 Banach Spaces
1.9 Notes
1.10 Exercises
1.11 Hints for the Exercises
2 Diagonal Operators
2.1 Diagonal Operators
2.2 Banach-Space Interlude
2.3 Inverse of an Operator
2.4 Spectrum of an Operator
2.5 Compact Diagonal Operators
2.6 Compact Selfadjoint Operators
2.7 Notes
2.8 Exercises
2.9 Hints for the Exercises
3 Infinite Matrices
3.1 Adjoint of an Operator
3.2 Special Case of Schur's Test
3.3 Schur's Test
3.4 Compactness and Contractions
3.5 Notes
3.6 Exercises
3.7 Hints for the Exercises
4 Two Multiplication Operators
4.1 Mx on L2[0, 1]
4.2 Fourier Analysis
4.3 Mξ on L2(T)
4.4 Notes
4.5 Exercises
4.6 Hints for the Exercises
5 The Unilateral Shift
5.1 The Shift on l2
5.2 Adjoint of the Shift
5.3 The Hardy Space
5.4 Bounded Analytic Functions
5.5 Multipliers of H2
5.6 Commutant of the Shift
5.7 Cyclic Vectors
5.8 Notes
5.9 Exercises
5.10 Hints for the Exercises
6 The Cesàro Operator
6.1 Cesàro Summability
6.2 The Cesàro Operator
6.3 Spectral Properties
6.4 Other Properties of the Cesàro Operator
6.5 Other Versions of the Cesàro Operator
6.6 Notes
6.7 Exercises
6.8 Hints for the Exercises
7 The Volterra Operator
7.1 Basic Facts
7.2 Norm, Spectrum, and Resolvent
7.3 Other Properties of the Volterra Operator
7.4 Invariant Subspaces
7.5 Commutant
7.6 Notes
7.7 Exercises
7.8 Hints for the Exercises
8 Multiplication Operators
8.1 Multipliers of Lebesgue Spaces
8.2 Cyclic Vectors
8.3 Commutant
8.4 Spectral Radius
8.5 Selfadjoint and Positive Operators
8.6 Continuous Functional Calculus
8.7 The Spectral Theorem
8.8 Revisiting Diagonal Operators
8.9 Notes
8.10 Exercises
8.11 Hints for the Exercises
9 The Dirichlet Shift
9.1 The Dirichlet Space
9.2 The Dirichlet Shift
9.3 The Dirichlet Shift is a 2-isometry
9.4 Multipliers and Commutant
9.5 Invariant Subspaces
9.6 Cyclic Vectors
9.7 The Bilateral Dirichlet Shift
9.8 Notes
9.9 Exercises
9.10 Hints for the Exercises
10 The Bergman Shift
10.1 The Bergman Space
10.2 The Bergman Shift
10.3 Invariant Subspaces
10.4 Invariant Subspaces of Higher Index
10.5 Multipliers and Commutant
10.6 Notes
10.7 Exercises
10.8 Hints for the Exercises
11 The Fourier Transform
11.1 The Fourier Transform on L1(R)
11.2 Convolution and Young's Inequality
11.3 Convolution and the Fourier Transform
11.4 The Poisson Kernel
11.5 The Fourier Inversion Formula
11.6 The Fourier–Plancherel Transform
11.7 Eigenvalues and Hermite Functions
11.8 The Hardy Space of the Upper Half-Plane
11.9 Notes
11.10 Exercises
11.11 Hints for the Exercises
12 The Hilbert Transform
12.1 The Poisson Integral on the Circle
12.2 The Hilbert Transform on the Circle
12.3 The Hilbert Transform on the Real Line
12.4 Notes
12.5 Exercises
12.6 Hints for the Exercises
13 Bishop Operators
13.1 The Invariant Subspace Problem
13.2 Lomonosov's Theorem
13.3 Universal Operators
13.4 Properties of Bishop Operators
13.5 Rational Case: Spectrum
13.6 Rational Case: Invariant Subspaces
13.7 Irrational Case
13.8 Notes
13.9 Exercises
13.10 Hints for the Exercises
14 Operator Matrices
14.1 Direct Sums of Hilbert Spaces
14.2 Block Operators
14.3 Invariant Subspaces
14.4 Inverses and Spectra
14.5 Idempotents
14.6 The Douglas Factorization Theorem
14.7 The Julia Operator of a Contraction
14.8 Parrott's Theorem
14.9 Polar Decomposition
14.10 Notes
14.11 Exercises
14.12 Hints for the Exercises
15 Constructions with the Shift Operator
15.1 The von Neumann–Wold Decomposition
15.2 The Sum of S and S*
15.3 The Direct Sum of S and S*
15.4 The Tensor Product of S and S*
15.5 Notes
15.6 Exercises
15.7 Hints for the Exercises
16 Toeplitz Operators
16.1 Toeplitz Matrices
16.2 The Riesz Projection
16.3 Toeplitz Operators
16.4 Selfadjoint and Compact Toeplitz Operators
16.5 The Brown–Halmos Characterization
16.6 Analytic and Co-analytic Symbols
16.7 Universal Toeplitz Operators
16.8 Notes
16.9 Exercises
16.10 Hints for the Exercises
17 Hankel Operators
17.1 The Hilbert Matrix
17.2 Doubly Infinite Hankel Matrices
17.3 Hankel Operators
17.4 The Norm of a Hankel Operator
17.5 Hilbert's Inequality
17.6 The Nehari Problem
17.7 The Carathéodory–Fejér Problem
17.8 The Nevanlinna–Pick Problem
17.9 Notes
17.10 Exercises
17.11 Hints for the Exercises
18 Composition Operators
18.1 A Motivating Example
18.2 Composition Operators on H2
18.3 Compact Composition Operators
18.4 Spectrum of a Composition Operator
18.5 Adjoint of a Composition Operator
18.6 Universal Operators and Composition Operators
18.7 Notes
18.8 Exercises
18.9 Hints for the Exercises
19 Subnormal Operators
19.1 Basics of Subnormal Operators
19.2 Cyclic Subnormal Operators
19.3 Subnormal Weighted Shifts
19.4 Invariant Subspaces
19.5 Notes
19.6 Exercises
19.7 Hints for the Exercises
20 The Compressed Shift
20.1 Model Spaces
20.2 From a Model Space to L2[0, 1]
20.3 The Compressed Shift
20.4 A Connection to the Volterra Operator
20.5 A Basis for the Model Space
20.6 A Matrix Representation
20.7 Notes
20.8 Exercises
20.9 Hints for the Exercises
References
Author Index
Subject Index
