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Toeplitz-like operators with rational symbol having poles on the unit circle

Jaftha, J.

2020

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Jaftha, J. (2020). Toeplitz-like operators with rational symbol having poles on the unit circle.

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Contents

1.1 Bounded Toeplitz operators . . . 2

1.1.1 Toeplitz operators: origins and applications . . . 2

1.1.2 Bounded multiplication and Toeplitz operators on Lp . . . 3

1.1.3 Shift invariance . . . 4

1.1.4 Fourier coefficients and the matrix representation . . . 5

1.1.5 Rational symbols . . . 7

1.1.6 Fredholm Theory . . . 8

1.1.7 The adjoint operator and selfadjoint Toeplitz operators . . 10

1.1.8 Matrix symbols . . . 11

1.2 Unbounded Toeplitz operators: L2 _{symbols and symbols in the} Smirnov class . . . 14

1.2.1 Unbounded Toeplitz operators on H2 with L2 symbols . . . 14

1.2.2 Unbounded Toeplitz operators on H2 with Smirnov class symbols . . . 15

1.3 Toeplitz-like operators with rational symbol having a pole on the unit circle . . . 16

1.3.1 Basic and Fredholm Properties . . . 17

1.3.2 The spectrum . . . 20

1.3.3 The adjoint operator and selfadjoint extensions . . . 21

1.3.4 Matrix symbols . . . 24

1.3.5 An example . . . 26

1.4 Outline of rest of the thesis . . . 29

Bibliography Chapter 1 . . . 30 xiii

2 Fredholm properties 35

2.1 Introduction . . . 35

2.2 Basic properties of Tω . . . 40

2.3 Intermezzo: Division with remainder by a polynomial in Hp _{. . . . 42}

2.4 Fredholm properties of Tωfor ω ∈ Rat(T) . . . 46

2.5 Fredholm properties of Tω: General case . . . 52

2.6 Matrix representation . . . 57

2.7 Examples . . . 61

Bibliography Chapter 2 . . . 64

3 The spectrum 67 3.1 Introduction . . . 67

3.2 Review and new results concerning Tω . . . 70

3.3 The spectrum of Tω . . . 74

3.4 The spectrum may be unbounded, the resolvent set empty . . . 76

3.5 The essential spectrum need not be connected . . . 78

3.6 A parametric example . . . 80

Bibliography Chapter 3 . . . 87

4 The adjoint 91 4.1 Introduction . . . 91

4.2 The operator Tω∗ for ω ∈ Rat(T) . . . 94

4.3 The adjoint of Tω for ω ∈ Rat(T) . . . 97

4.4 The adjoint of Tω: General case . . . 101

4.5 Symmetric operators and selfadjoint extensions . . . 103

4.6 Comparison with the unbounded Toeplitz operator defined by Sarason109 Bibliography Chapter 4 . . . 111

5 Matrix symbols 115 5.1 Introduction . . . 115

5.2 Basic properties of TΩ . . . 118

5.3 Matrix Function Factorization . . . 122

5.4 An example . . . 127

5.5 Factorization of the Toeplitz operator . . . 129

5.6 Fredholm properties . . . 131

Bibliography Chapter 5 . . . 136