The Role of the Spectrum in the Cyclic

Introduction and preliminaries Linear fractional maps with an interior fixed point Non elliptic automorphisms The parabolic non automorphism Supercyclic linear fractional composition operators Endnotes Bibliography

Preface

ix

Chapter 1. Introduction and Preliminaries

1

(12)

Weighted Hardy spaces

1

(1)

The Dirichlet space

2

(1)

The Hardy space

2

(1)

Functional Hilbert spaces

2

(1)

Composition operators

3

(1)

Weighted Dirichlet spaces

3

(1)

Equivalent norms on weighted Dirichlet spaces

3

(1)

Linear fractional maps

4

(1)

Iterates

5

(1)

The Comparison Principle

6

(1)

Main results

7

(6)

Table I

8

(2)

Table II

10

(2)

Acknowledgment

12

(1)

Chapter 2. Linear Fractional Maps with an Interior Fixed Point

13

(22)

Non-elliptic linear fractional maps with an exterior fixed point

13

(4)

Elliptic automorphism

17

(2)

Interior and boundary fixed point

19

(4)

Hypercyclicity

23

(12)

The hyperbolic automorphism

24

(1)

The Hypercyclic Criterion

24

(1)

A Beurling type theorem

25

(1)

The Inner Product Criterion

25

(6)

Invariant subspaces of finite codimension

31

(4)

Chapter 3. Non Elliptic Automorphisms

35

(13)

Hypercyclicity

34

(6)

The parabolic automorphism

36

(2)

The hyperbolic automorphism

38

(2)

The failure of the spectrum

40

(3)

The eigenfunctions of Cφ with φ a parabolic automorphism

42

(1)

Cyclicity

43

(5)

The Dirichlet space of the upper half plane

44

(1)

The Hardy space of the upper half plane

44

(1)

A Paley-Wiener Theorem for Dπ

44

(1)

The parabolic automorphism

45

(1)

The hyperbolic automorphism

46

(2)

Chapter 4. The Parabolic Non Automorphism

48

(18)

Cyclicity

48

(4)

Zeros of weighted Dirichlet functions

51

(1)

Non hypercyclicity

52

(14)

An appetizer

52

(1)

The spectrum of Cφ

53

(1)

The eigenfunctions form a spanning set

53

(1)

Renorming Sν

54

(1)

Isometric functional Hilbert spaces

55

(1)

A complete orthogonal system of L² (R+, dt/t)

56

(3)

The extension of the isometrics

59

(1)

The Laguerre polynomials

60

(6)

Chapter 5. Supercyclic Linear Fractional Composition Operators

66

(11)

The easy part

66

(2)

The parabolic non-automorphism

68

(9)

The Angle Criterion

68

(1)

The adjoint of Cφ

69

(2)

A three term recurrence formula

71

(1)

An integration formula

71

(6)

Endnotes

77

(2)

Bibliography

79

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The Role of the Spectrum in the Cyclic Behavior of Composition Operators

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