Cover

Title page

Contents

Preface

Approximation numbers of composition operators on a Hilbert space of Dirichlet series

1. Introduction and statement of main results

2. The space \Ht of Dirichlet series

3. Bounded composition operators on \Ht

4. Definitions and tools from operator theory

5. Definitions and tools from function theory

6. General estimates for approximation numbers

7. Statement of the main results

8. Proof of one theorem

9. Acknowledgement

References

A short introduction to de Branges–Rovnyak spaces

1. Introduction

2. Preliminaries

3. Introducing de Branges–Rovnyak spaces

4. More about contractively included subspaces

5. Back to \HH(𝑏)

6. The nonextreme case

7. The extreme case

8. \HH(𝑏) as a model space

9. Further reading

References

Asymptotic Bohr radius for the polynomials in one complex variable

1. Introduction

2. Main Theorem

References

A survey on preservers of spectra and local spectra

1. Introduction

2. Our approach

3. Basic notations

4. Various forms of Kaplansky’s problem

4.1. Kaplansky’s conjecture

4.2. Preservers on matrices

4.3. Preservers of operators

4.4. Preservers of spectrum

4.5. Preservers on Banach algebras

4.6. Spectral isometries and spectrally bounded maps

4.7. Left and right invertibility preservers

5. Linear maps preserving semi-Fredholm operators and generalized invertibility

5.1. Linear maps preserving Generalized invertibility, and semi-Fredholm operators

5.2. Linear maps preserving Fredholm and Atkinson elements of Banach algebras

5.3. Linear maps preserving the essential spectral radius

6. Minimum, surjectivity and reduced minimum moduli preservers

6.1. Minimum and surjectivity moduli preservers

6.2. Reduced minimum modulus preservers

6.3. The surjectivity and inner spectral radii preservers

6.4. Minimum, surjectivity and reduced minimum moduli preservers in 𝐶*-algebras

7. Local spectra preservers

7.1. Background from local spectral theory

7.2. Linear preservers of local spectrum

7.3. Inner local spectral radius and preservers

7.4. Outer local spectral radius and preservers

7.5. Nonlinear preservers of local spectrum

7.6. Preservers of local spectra at non fixed vectors

References

Commutants of finite Blaschke product multiplication operators

1. Introduction

2. General Theory

3. Relations between Finite Blaschke Products and the Structure of ℋ

4. Commutant of 𝑇_{𝐵} for 𝐵 a Finite Blaschke Product

5. A Special Annulus and the Wrapping Function

6. Consequences for Commutants of Operators on \Htbk

7. A More Detailed Description of Operators in the Commutant

References

Complex approximation and extension-interpolation on arbitrary sets in one dimension

1. Rational approximation

2. Extension-interpolation

References

Cyclicity in non-extreme de Branges-Rovnyak spaces

1. Introduction

2. Cyclicity for 𝑋_{𝑏}

3. Cyclicity for 𝑌_{𝑏}

4. An important example

References

Integral representations of the derivatives in ℋ(𝒷) spaces

1. Introduction

2. Preliminaries

3. Derivatives of Blaschke products

4. Higher derivatives of 𝑏

5. Approximation by Blaschke products

6. Reproducing kernels for derivatives

7. An interpolation problem

8. Derivatives of \HH(𝑏) functions

9. Passage to the upper half plane \C₊

10. Integral representations for derivatives

References

Interpolation and moment in weighted Hardy spaces

1. Introduction

2. Expansions in 𝐻^{𝑝}_{+𝑤}

3. Interpolation and expansion

4. Moment and expansion

References

Model spaces: A survey

1. Introduction

2. Preliminaries

3. Model spaces

4. Model operators

5. Reproducing kernels

6. Boundary behavior

7. Conjugation

8. Aleksandrov-Clark measures

9. Completeness problems

10. Model spaces for the upper-half plane

11. Generalizations of model spaces

12. Truncated Toeplitz operators

13. Things we did not mention

References

Note on a Julia operator related to model spaces

1. Introduction

2. Preliminaries

3. A unitary operator

4. Various roles of Ω

Acknowledgements

References

Selected problems in classical function theory

1. Introduction: Spaces of Analytic Functions

2. Analytic Functions with Positive Boundary Values

3. Analytic Content

4. Putnam’s Inequality for Toeplitz Operators in Bergman Spaces

References

The linear bound for Haar multiplier paraproducts

1. Introduction and Statement of Main Results

2. Notation and Useful Facts

3. Linear Bound for Haar Multipliers

References

Transitivity and bundle shifts

1. Introduction

2. Function Algebras

3. Catalytic in Function algebras

4. Finitely-connected domains

References

Weak 𝐻¹, the real and complex case

1. The topology of the Lorentz 𝐿(𝑝,𝑞) spaces

2. The real weak 𝐻¹(ℝⁿ)space

3. The Complex Case

References

Back Cover