**Contemporary Mathematics**

Volume: 638;
2015;
317 pp;
Softcover

MSC: Primary 47; 30; 31; 32;

**Print ISBN: 978-1-4704-1045-2
Product Code: CONM/638**

List Price: $112.00

AMS Member Price: $89.60

MAA Member Price: $100.80

**Electronic ISBN: 978-1-4704-2341-4
Product Code: CONM/638.E**

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# Invariant Subspaces of the Shift Operator

Share this page *Edited by *
*Javad Mashreghi; Emmanuel Fricain; William Ross*

A co-publication of the AMS and Centre de Recherches Mathématiques

This volume contains the proceedings of the CRM Workshop on Invariant
Subspaces of the Shift Operator, held August 26–30, 2013, at the
Centre de Recherches Mathématiques, Université de Montréal,
Montréal, Quebec, Canada.

The main theme of this volume is the invariant subspaces of the
shift operator (or its adjoint) on certain function spaces, in
particular, the Hardy space, Dirichlet space, and de
Branges–Rovnyak spaces.

These spaces, and the action of the shift operator on them, have
turned out to be a precious tool in various questions in analysis such
as function theory (Bieberbach conjecture, rigid functions,
Schwarz–Pick inequalities), operator theory (invariant subspace
problem, composition operator), and systems and control theory.

Of particular interest is the Dirichlet space, which is one of the
classical Hilbert spaces of holomorphic functions on the unit
disk. From many points of view, the Dirichlet space is an interesting
and challenging example of a function space. Though much is known
about it, several important open problems remain, most notably the
characterization of its zero sets and of its shift-invariant
subspaces.

#### Readership

Graduate students and research mathematicians interested in operator theory and function spaces.

# Table of Contents

## Invariant Subspaces of the Shift Operator

- Cover Cover11 free
- Title page i2 free
- Contents v6 free
- Preface vii8 free
- Approximation numbers of composition operators on a Hilbert space of Dirichlet series 110 free
- 1. Introduction and statement of main results 110
- 2. The space \Ht of Dirichlet series 211
- 3. Bounded composition operators on \Ht 312
- 4. Definitions and tools from operator theory 312
- 5. Definitions and tools from function theory 514
- 6. General estimates for approximation numbers 716
- 7. Statement of the main results 918
- 8. Proof of one theorem 1524
- 9. Acknowledgement 1726
- References 1726

- A short introduction to de Branges–Rovnyak spaces 2130
- Asymptotic Bohr radius for the polynomials in one complex variable 3948
- A survey on preservers of spectra and local spectra 4554
- 1. Introduction 4655
- 2. Our approach 4756
- 3. Basic notations 4857
- 4. Various forms of Kaplansky’s problem 4958
- 5. Linear maps preserving semi-Fredholm operators and generalized invertibility 6675
- 6. Minimum, surjectivity and reduced minimum moduli preservers 7180
- 7. Local spectra preservers 7786
- 7.1. Background from local spectral theory 7887
- 7.2. Linear preservers of local spectrum 8089
- 7.3. Inner local spectral radius and preservers 8392
- 7.4. Outer local spectral radius and preservers 8493
- 7.5. Nonlinear preservers of local spectrum 8594
- 7.6. Preservers of local spectra at non fixed vectors 8695

- References 8897

- Commutants of finite Blaschke product multiplication operators 99108
- 1. Introduction 99108
- 2. General Theory 101110
- 3. Relations between Finite Blaschke Products and the Structure of ℋ 103112
- 4. Commutant of 𝑇_{𝐵} for 𝐵 a Finite Blaschke Product 105114
- 5. A Special Annulus and the Wrapping Function 107116
- 6. Consequences for Commutants of Operators on \Htbk 108117
- 7. A More Detailed Description of Operators in the Commutant 111120
- References 113122

- Complex approximation and extension-interpolation on arbitrary sets in one dimension 115124
- Cyclicity in non-extreme de Branges-Rovnyak spaces 131140
- Integral representations of the derivatives in ℋ(𝒷) spaces 137146
- 1. Introduction 137146
- 2. Preliminaries 139148
- 3. Derivatives of Blaschke products 140149
- 4. Higher derivatives of 𝑏 145154
- 5. Approximation by Blaschke products 149158
- 6. Reproducing kernels for derivatives 154163
- 7. An interpolation problem 157166
- 8. Derivatives of \HH(𝑏) functions 163172
- 9. Passage to the upper half plane \C₊ 168177
- 10. Integral representations for derivatives 170179
- References 177186

- Interpolation and moment in weighted Hardy spaces 179188
- Model spaces: A survey 197206
- 1. Introduction 197206
- 2. Preliminaries 198207
- 3. Model spaces 205214
- 4. Model operators 209218
- 5. Reproducing kernels 212221
- 6. Boundary behavior 217226
- 7. Conjugation 220229
- 8. Aleksandrov-Clark measures 225234
- 9. Completeness problems 230239
- 10. Model spaces for the upper-half plane 232241
- 11. Generalizations of model spaces 236245
- 12. Truncated Toeplitz operators 238247
- 13. Things we did not mention 240249
- References 241250

- Note on a Julia operator related to model spaces 247256
- Selected problems in classical function theory 255264
- The linear bound for Haar multiplier paraproducts 267276
- Transitivity and bundle shifts 287296
- Weak 𝐻¹, the real and complex case 299308
- Back Cover Back Cover1330