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E-raamat: Modern Approaches to the Invariant-Subspace Problem

Isabelle Chalendar (Université Lyon I), Jonathan R. Partington (University of Leeds)

Formaat: PDF+DRM
Sari: Cambridge Tracts in Mathematics
Ilmumisaeg: 18-Aug-2011
Kirjastus: Cambridge University Press
Keel: eng
ISBN-13: 9781139120128

Teised raamatud teemal:

Functional analysis & transforms

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Formaat: PDF+DRM
Sari: Cambridge Tracts in Mathematics
Ilmumisaeg: 18-Aug-2011
Kirjastus: Cambridge University Press
Keel: eng
ISBN-13: 9781139120128

Teised raamatud teemal:

Functional analysis & transforms

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"One of the major unsolved problems in operator theory is the fifty-year-old invariant subspace problem, which asks whether every bounded linear operator on a Hilbert space has a nontrivial closed invariant subspace. This book presents some of the major results in the area, including many that were derived within the past few years and cannot be found in other books. Beginning with a preliminary chapter containing the necessary pure mathematical background, the authors present a variety of powerful techniques, including the use of the operator-valued Poisson kernel, various forms of the functional calculus, Hardy spaces, fixed point theorems, minimal vectors, universal operators and moment sequences. The subject is presented at a level accessible to postgraduate students, as well as established researchers. It will be of particular interest to those who study linear operators and also to those who work in other areas of pure mathematics"--

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Arvustused

'I think this is a very useful book which will serve as a good source for a rich variety of methods that have been developed for proving positive results on the ISP. Moreover, there is much material in the book which is of interest beyond its application to the ISP. [ It] should be of interest to analysts in general as well as being an essential source for study of the ISP.' Sandy Davie, SIAM Review

Muu info

Presents work on the invariant subspace problem, a major unsolved problem in operator theory.

Preface

1 Background

(36)

1.1 Functional analysis

(13)

1.1.1 Weak topology

(2)

1.1.2 Hahn-Banach theorem

(1)

1.1.3 Stone-Weierstrass theorem

(1)

1.1.4 Banach-Steinhaus theorem

(1)

1.1.5 Complex measures

(5)

1.1.6 Riesz representation theorem

(2)

1.1.7 Geometry of Banach spaces

(1)

1.2 Operator theory

(8)

1.2.1 Basic definitions and spectral properties

(6)

1.2.2 Wold decomposition of an isometry

(1)

1.2.3 Riesz-Dunford functional calculus

(1)

1.3 The Poisson kernel

(1)

1.4 Hardy spaces

(13)

1.4.1 Inner and outer functions

(3)

1.4.2 Consequences of the inner-outer factorization

(2)

1.4.3 The theorems of Beurling and Wiener

(1)

1.4.4 The disc algebra

(1)

1.4.5 Reproducing kernels, Riesz bases and Carleson sequences

(3)

1.4.6 Functions of bounded mean oscillation

(1)

1.4.7 The Hilbert transform on the unit circle

(1)

1.5 Number Theory

(1)

2 The operator-valued Poisson kernel and its applications

(20)

2.1 The operator-valued Poisson kernel

(6)

2.2 The H∞ functional calculus for absolutely continuous p-contractions

(3)

2.3 H∞ functional calculus in a complex Banach space

(4)

2.4 Absolutely continuous elementary spectral measures

(7)

Exercises

(1)

Comments

(3)

3 Properties (An,m) and factorization of integrable functions

(46)

3.1 The basis of the S. Brown method

(10)

3.1.1 The starting point

(5)

3.1.2 The class A

(1)

3.1.3 Classes An,m

(4)

3.2 Factorization of log-integrable functions

(14)

3.3 Applications in harmonic analysis

(5)

3.4 Subnormal operators

(6)

3.4.1 Borelian functional calculus for normal operators

(1)

3.4.2 Invariant subspaces for subnormal operators

(5)

3.5 Surjectivity of continuous bilinear mapping

(11)

3.5.1 A sufficient condition for property (A0)

(4)

3.5.2 A sufficient condition for property (A1,0)

(3)

Exercises

(1)

Comments

100

(3)

4 Polynomially bounded operators with rich spectrum

103

(38)

4.1 Apostol's theorem

103

(4)

4.2 C2(T) functional calculus and the Colojoara-Foias theorem

107

(4)

4.2.1 Operators with a C2(T) functional calculus

107

(3)

4.2.2 The Colojoara-Foias, theorem

110

(1)

4.3 Zenger's theorem

111

(7)

4.3.1 Zenger's theorem and a factorization result

112

(2)

4.3.2 A stronger version of Zenger's theorem

114

(4)

4.4 Carleson's interpolation theorem

118

(5)

4.5 Approximation using Apostol sets

123

(6)

4.5.1 Approximation of integrable non-negative functions

123

(5)

4.5.2 Approximate eigenvalues

128

(1)

4.6 Invariant subspace results

129

(12)

Exercises

137

(1)

Comments

138

(3)

5 Beurling algebras

141

(28)

5.1 Properties of Beurling algebras

142

(4)

5.2 Theorems of Wermer and Atzmon

146

(6)

5.3 Bishop operators

152

(8)

5.3.1 Davie's functional calculus

152

(4)

5.3.2 The point spectrum

156

(4)

5.4 Rational Bishop operators

160

(9)

5.4.1 Cyclic vectors

161

(2)

5.4.2 The lattice of invariant subspaces

163

(4)

Exercises

167

(1)

Comments

167

(2)

6 Applications of a fixed-point theorem

169

(14)

6.1 Operators commuting with compact operators

169

(2)

6.2 Essentially self-adjoint operators

171

(12)

6.2.1 Preliminaries

171

(6)

6.2.2 Application to invariant subspaces

177

(3)

Exercises

180

(1)

Comments

181

(2)

7 Minimal vectors

183

(30)

7.1 The basic definitions

183

(2)

7.2 Minimal vectors in Hilbert space

185

(1)

7.3 A general extremal problem

186

(6)

7.3.1 Approximation in Hilbert spaces

187

(2)

7.3.2 Approximation in reflexive Banach spaces

189

(3)

7.4 Application to hyperinvariant subspaces

192

(21)

7.4.1 The main theorem

192

(3)

7.4.2 Compact operators

195

(1)

7.4.3 Weighted composition operators

196

(9)

7.4.4 Weighted shifts

205

(3)

7.4.5 Multiplication operators on Lp spaces

208

(3)

Exercises

211

(1)

Comments

211

(2)

8 Universal operators

213

(20)

8.1 Construction of universal models

213

(4)

8.2 Bilateral weighted shifts

217

(3)

8.3 Composition operators

220

(13)

8.3.1 Universality of composition operators

220

(4)

8.3.2 Minimal subspaces and eigenfunctions

224

(4)

Exercises

228

(2)

Comments

230

(3)

9 Moment sequences and binomial sums

233

(22)

9.1 Moment sequences

233

(6)

9.2 Operators on sequence spaces

239

(2)

9.3 Binomial sums

241

(14)

9.3.1 Proof of Theorem 9.3.1

242

(2)

9.3.2 A technical refinement

244

(4)

9.3.3 Application to Banach algebras and invariant subspaces

248

(3)

Exercises

251

(1)

Comments

252

(3)

10 Positive and strictly-singular operators

255

(14)

10.1 Ordered spaces and positive operators

255

(2)

10.2 Invariant subspaces for positive operators

257

(6)

10.3 Strictly singular operators

263

(6)

Exercises

265

(1)

Comments

266

(3)

References

269

(12)

Index

281

Isabelle Chalendar is an Assistant Professor in the Department of Mathematics at the University of Lyon 1, France. Jonathan R. Partington is a Professor in the School of Mathematics at the University of Leeds.

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