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

by Gallardo-Gutierrez, Eva A.; Montes-Rodriguez, Alfonso

• ISBN13: 9780821834329
• ISBN10: 0821834320
• Format: Paperback
• Copyright: 2004-01-01
• Publisher: Amer Mathematical Society

Summary

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

Table of Contents

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