Operator Theory by Example

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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