**Contemporary Mathematics**

Volume: 561;
2012;
252 pp;
Softcover

MSC: Primary 30; 47;

**Print ISBN: 978-0-8218-5275-0
Product Code: CONM/561**

List Price: $95.00

AMS Member Price: $76.00

MAA Member Price: $85.50

**Electronic ISBN: 978-0-8218-8495-9
Product Code: CONM/561.E**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

# Topics in Complex Analysis and Operator Theory

Share this page *Edited by *
*Óscar Blasco; José A. Bonet; José M. Calabuig; David Jornet*

This book contains the lecture notes as well as some invited papers
presented at the Third Winter School in Complex Analysis, Operator
Theory and Applications held February 2–5, 2010, in Valencia,
Spain.

The book is divided into two parts. The first is an extended
self-contained version of the mini-courses taught at the School. The
papers in this first part are: Notes on real analytic functions
and classical operators, by Paweł Domański;
Shining a Hilbertian lamp on the bidisk, by John E. McCarthy;
Selected problems in perturbation theory, by Vladimir
V. Peller; and Composition operators on Hardy–Orlicz
spaces, by Luis Rodríguez-Piazza.

The second part consists of several research papers on recent
advances in the area and some survey articles of an expository
character. The articles in this second part are: Remarks on weighted
mixed norm spaces, by O. Blasco; Interpolation subspaces of
\(L^1\) of a vector measure and norm inequalities for the
integration operator, by J.M. Calabuig, J. Rodríguez, and
E.A. Sánchez-Pérez; On the spectra of algebras of
analytic functions, by D. Carando, D. García, M. Maestre,
and P. Sevilla-Peris; Holomorphic self-maps of the disk
intertwining two linear fractional maps, by M.D. Contreras,
S. Díaz-Madrigal, M.J. Martín, and D. Vukotić;
ABC-type estimates via Garsia-type norms, by K.M. Dyakonov;
and Volterra type operators on Bergman spaces with exponential
weights, by J. Pau and J.A. Peláez.

The topics selected for the mini-courses cover several aspects of
complex analysis and operator theory that play important roles in
understanding connections between different areas that are
considered in fashion these days. This part is aimed at graduate
students and young researchers. The courses are self-contained,
focusing on those aspects that are basic and that can lead the readers
to a quick understanding of the theories presented in each topic. They
start with the classical results and reach a selection of open
problems in each case. The research and survey articles are aimed at
young researchers in the area, as well as post-doc and senior
researchers interested in complex analysis and operator
theory.

This book is published in cooperation with Real Sociedad Matemática Española (RSME)

#### Readership

Graduate students and research mathematicians interested in complex analysis and operator theory.

# Table of Contents

## Topics in Complex Analysis and Operator Theory

- Preface vii8 free
- List of Participants ix10 free
- Mini-courses 114 free
- Notes on Real Analytic Functions and Classical Operators 316
- 1. Lecture —Operator relevant properties of the space of real analytic functions 417
- 2. Lecture 2—Composition operators on the space of real analytic functions 1629
- 3. Lecture 3—Differential and convolution operators on the space of real analytic functions 2639
- 4. Lecture 4—Isomorphism of the spaces of real analytic functions 3447
- References 4255

- Shining a Hilbertian lamp on the bidisk 4962
- Selected problems in perturbation theory 6780
- 1. Introduction 6780
- 2. Double operator integrals 6982
- 3. Besov spaces 7285
- 4. Nuclearity of Hankel operators 7487
- 5. Operator Lipschitz and operator differentiable functions. Sufficient conditions 7487
- 6. Operator Lipschitz and operator differentiable functions. Necessary conditions 7992
- 7. Operator Hölder–Zygmund functions 8194
- 8. Lifshits–Krein trace formulae 8396
- 9. Koplienko–Neidhardt trace formulae 8598
- 10. Perturbations of class 𝐒_{𝐩} 8598
- 11. Perturbations of normal operators 88101
- References 89102

- Composition operators on Hardy-Orlicz spaces 91104
- Articles 135148
- Remarks on weighted mixed norm spaces 137150
- 1. Introduction and preliminaries 137150
- 2. Generalities on weights. 139152
- 3. Generalities on weighted mixed norm spaces. 142155
- 4. Bergman type projections on weighted mixed norm spaces. 145158
- 5. Continuity of the Berezin transform on weighted mixed norm spaces. 147160
- 6. The averaging operator on weighted mixed norm spaces. 150163
- References 153166

- Interpolation subspaces of 𝐿¹ of a vector measure and norm inequalities for the integration operator 155168
- On the spectra of algebras of analytic functions 165178
- 1. Introduction 165178
- 2. The Corona Theorem for 𝐻^{∞}(𝐵) 167180
- 3. A weak Corona theorem: The Cluster Value Theorem for 𝐻^{∞}(𝐵) and 𝐴ᵤ(𝐵) 170183
- 4. The size of the fibers and cluster sets for some algebras of bounded analytic functions 174187
- 5. The spectrum of 𝐻_{𝑏}(𝑈) 176189
- 6. Weighted algebras of holomorphic functions 182195
- 7. Homomorphisms between algebras 187200
- References 195208

- Holomorphic self-maps of the disk intertwining two linear fractional maps 199212
- 𝐴𝐵𝐶-type estimates via Garsia-type norms 229242
- Volterra type operators on Bergman spaces with exponential weights 239252