An Advanced Complex Analysis Problem Book: Topological Vector Spaces, Functional Analysis, and Hilbert Spaces of Analytic Functions

523 Pages·2015·3.1 MB·English

by  Daniel Alpay

Daniel Alpay
An Advanced
Complex Analysis
Problem Book
Topological Vector Spaces,
Functional Analysis, and
Hilbert Spaces of Analytic Functions

Daniel Alpay
An Advanced
Complex Analysis
Problem Book
Topological Vector Spaces, Functional Analysis,
and Hilbert Spaces of Analytic Functions
Daniel Alpay
Ben-Gurion University of the Negev
Beer Sheva, Israel
ISBN 978-3-319-16058-0 ISBN 978-3-319-16059-7 (eBook)
DOI 10.1007/978-3-319-16059-7

Library of Congress Control Number: 2015956341
Mathematics Subject Classification (2010): 97I80, 46C07, 46E20, 46E22, 47B32
S pringer Cham Heidelberg New York Dordrecht London
© Springer International Publishing Switzerland 2015
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Springer International Publishing AG Switzerland is part of Springer Science+Business Media
(www.birkhauser-science.com)
It is a pleasure to dedicate this work to my son Gabriel. It is also a
pleasure to thank the Earl Katz family for endowing the chair (Earl
Katz Family Chair in Algebraic System Theory), which supported this
research.

Contents
I Analytic Functions 9
1 Algebraic Prerequisites 11
1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.6 Positive matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.7 Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.8 Solutions of the exercises. . . . . . . . . . . . . . . . . . . . . . . . 42
2 Analytic Functions 61
2.1 Some warm-up exercises . . . . . . . . . . . . . . . . . . . . . . . . 61
2.2 Some important theorems to recall . . . . . . . . . . . . . . . . . . 75
2.3 Cauchy’s formula and Laurent expansion. . . . . . . . . . . . . . . 77
2.4 Star-shaped and simply connected sets . . . . . . . . . . . . . . . . 82
2.5 Various . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.6 Harmonic and subharmonic functions. . . . . . . . . . . . . . . . . 85
2.7 Meromorphic functions in C . . . . . . . . . . . . . . . . . . . . . . 89
2.8 Solutions of the exercises. . . . . . . . . . . . . . . . . . . . . . . . 90
II Topology and Functional Analysis 119
3 Topological Spaces 121
3.1 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.2 Compact sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.3 Connected sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.4 Continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.5 Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.6 Closed, open and proper maps . . . . . . . . . . . . . . . . . . . . 137
3.7 Quotient topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
vii
viii Contents
3.8 Manifolds and surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 142
3.9 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
3.10 Solutions of the exercises. . . . . . . . . . . . . . . . . . . . . . . . 157
4 Normed Spaces 199
4.1 Normed Banach and Hilbert spaces . . . . . . . . . . . . . . . . . . 199
4.2 Operators on normed spaces . . . . . . . . . . . . . . . . . . . . . . 207
4.3 Unbounded operators . . . . . . . . . . . . . . . . . . . . . . . . . 222
4.4 Indefinite inner product spaces . . . . . . . . . . . . . . . . . . . . 225
4.5 Solutions of the exercises. . . . . . . . . . . . . . . . . . . . . . . . 227
5 Locally Convex Topological Vector Spaces 249
5.1 Topological vector spaces . . . . . . . . . . . . . . . . . . . . . . . 249
5.2 Countably normed spaces and Fr´echet spaces . . . . . . . . . . . . 253
5.3 Operators in countably normed spaces . . . . . . . . . . . . . . . . 257
5.4 Dual of a Fr´echet space . . . . . . . . . . . . . . . . . . . . . . . . 259
5.5 Positive operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
5.6 Topology of the space of analytic functions . . . . . . . . . . . . . 264
5.7 Normal families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
5.8 The dual of the space of analytic functions. . . . . . . . . . . . . . 268
5.9 Solutions of the exercises. . . . . . . . . . . . . . . . . . . . . . . . 268
6 Some Functional Analysis 285
6.1 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
6.2 Stieltjes integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
6.3 Density results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
6.4 Solutions of the exercises. . . . . . . . . . . . . . . . . . . . . . . . 306
III Hilbert Spaces of Analytic Functions 329
7 Reproducing Kernel Hilbert Spaces 331
7.1 Positive definite kernels . . . . . . . . . . . . . . . . . . . . . . . . 332
7.2 Examples of positive definite functions and kernels . . . . . . . . . 338
7.3 Conditionally negative functions . . . . . . . . . . . . . . . . . . . 347
7.4 Vector-valued functions . . . . . . . . . . . . . . . . . . . . . . . . 350
7.5 Reproducing kernel Hilbert spaces . . . . . . . . . . . . . . . . . . 352
7.6 Linear operators in reproducing kernel Hilbert spaces. . . . . . . . 361
7.7 Finite-dimensional reproducing kernel spaces . . . . . . . . . . . . 367
7.8 Solutions of the exercises. . . . . . . . . . . . . . . . . . . . . . . . 376
Contents ix
8 Hardy Spaces 405
8.1 Reproducing kernel Hilbert spaces of analytic functions . . . . . . 405
8.2 The Hardy space of the open unit disk . . . . . . . . . . . . . . . . 406
8.3 Some operator theory in H2(D) . . . . . . . . . . . . . . . . . . . . 409
8.4 Composition operators . . . . . . . . . . . . . . . . . . . . . . . . . 412
8.5 Cuntz relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
8.6 The Hardy space of the open upper half-plane . . . . . . . . . . . . 415
8.7 The fractional Hardy space Hν of the open
2
upper half-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
8.8 Solutions of the exercises. . . . . . . . . . . . . . . . . . . . . . . . 421
9 de Branges–RovnyakSpaces 443
9.1 de Branges–Rovnyakspaces . . . . . . . . . . . . . . . . . . . . . . 443
9.2 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
9.3 Carath´eodory’stheorem . . . . . . . . . . . . . . . . . . . . . . . . 449
9.4 A few words on Schur analysis . . . . . . . . . . . . . . . . . . . . 450
9.5 Solutions of the exercises. . . . . . . . . . . . . . . . . . . . . . . . 453
10 Bergman Spaces 461
10.1 The Bergman space of analytic functions analytic in D . . . . . . . 461
10.2 The Bergman space of analytic functions analytic in an ellipse. . . 463
10.3 The Bergman space of the annulus . . . . . . . . . . . . . . . . . . 464
10.4 The Bergman spaces of polyanalytic functions . . . . . . . . . . . . 464
10.5 Solutions of the exercises. . . . . . . . . . . . . . . . . . . . . . . . 466
11 Fock Spaces 475
11.1 The Bargmann–Fock–Segalspaces of analytic functions . . . . . . 475
11.2 The Bargmann–Fock–Segalspaces of
polyanalytic functions . . . . . . . . . . . . . . . . . . . . . . . . . 478
11.3 Solutions of the exercises. . . . . . . . . . . . . . . . . . . . . . . . 479
Bibliography 487
Index 509
Name Index 518
Notation Index 521