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E-raamat: Theory of H(b) Spaces: Volume 1

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  • Sari: New Mathematical Monographs
  • Ilmumisaeg: 26-May-2016
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781316053812
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Theory of H(b) Spaces: Volume 1Suurem pilt
  • Formaat: EPUB+DRM
  • Sari: New Mathematical Monographs
  • Ilmumisaeg: 26-May-2016
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781316053812
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"An H(b) space is defined as a collection of analytic functions which are in the image of an operator. The theory of H(b) spaces bridges two classical subjects: complex analysis and operator theory, which makes it both appealing and demanding. The first volume of this comprehensive treatment is devoted to the preliminary subjects required to understand the foundation of H(b) spaces, such as Hardy spaces, Fourier analysis, integral representation theorems, Carleson measures, Toeplitz and Hankel operators,various types of shift operators, and Clark measures. The second volume focuses on the central theory. Both books are accessible to graduate students as well as researchers: each volume contains numerous exercises and hints, and figures are included throughout to illustrate the theory. Together, these two volumes provide everything the reader needs to understand and appreciate this beautiful branch of mathematics"--

In two volumes, this comprehensive treatment covers all that is needed to understand and appreciate this beautiful branch of mathematics.

Arvustused

'This two volume monograph is a compendium of the H(b) spaces that will be of interest to both graduate students and practicing mathematicians interested in function-theoretic operator theory. There are 31 chapters between the two volumes and a detailed bibliography consisting of 766 entries. The first volume is devoted to general function-theoretic operator theory (and indeed is a useful reference in its own right) while the second volume is more specialized and contains an in-depth survey of H(b)H(b) theory and related ideas.' Steve Deckelman, MAA Reviews ' designed for a person who wants to learn the theory of these spaces and understand the state of the art in the area. All major results are included. In some situations the original proofs are provided, while in other cases they provide the 'better' proofs that have become available since. The books are designed to be accessible to both experts and newcomers to the area. Comments at the end of each section are very helpful, and the numerous exercises were clearly chosen to help master some of the techniques and tools used In sum, these are excellent books that are bound to become standard references for the theory of H(b) spaces.' Bulletin of the American Mathematical Society

Muu info

In two volumes, this comprehensive treatment covers all that is needed to understand and appreciate this beautiful branch of mathematics.
Preface xvii
1 Normed linear spaces and their operators
1(59)
1.1 Banach spaces
1(8)
1.2 Bounded operators
9(5)
1.3 Fourier series
14(1)
1.4 The Hahn--Banach theorem
15(6)
1.5 The Baire category theorem and its consequences
21(5)
1.6 The spectrum
26(4)
1.7 Hilbert space and projections
30(10)
1.8 The adjoint operator
40(5)
1.9 Tensor product and algebraic direct sum
45(4)
1.10 Invariant subspaces and cyclic vectors
49(3)
1.11 Compressions and dilations
52(2)
1.12 Angle between two subspaces
54(6)
Notes on
Chapter 1
57(3)
2 Some families of operators
60(36)
2.1 Finite-rank operators
60(2)
2.2 Compact operators
62(3)
2.3 Subdivisions of spectrum
65(5)
2.4 Self-adjoint operators
70(7)
2.5 Contractions
77(1)
2.6 Normal and unitary operators
78(2)
2.7 Forward and backward shift operators on ll2
80(3)
2.8 The multiplication operator on L2(μ)
83(3)
2.9 Doubly infinite Toeplitz and Hankel matrices
86(10)
Notes on
Chapter 2
92(4)
3 Harmonic functions on the open unit disk
96(26)
3.1 Nontangential boundary values
96(2)
3.2 Angular derivatives
98(3)
3.3 Some well-known facts in measure theory
101(5)
3.4 Boundary behavior of Pμ
106(4)
3.5 Integral means of Pμ
110(2)
3.6 Boundary behavior of Qμ
112(1)
3.7 Integral means of Qμ
113(3)
3.8 Subharmonic functions
116(1)
3.9 Some applications of Green's formula
117(5)
Notes on
Chapter 3
120(2)
4 Hardy spaces
122(44)
4.1 Hyperbolic geometry
122(2)
4.2 Classic Hardy spaces Hp
124(6)
4.3 The Riesz projection P+
130(5)
4.4 Kernels of P+ and P-
135(2)
4.5 Dual and predual of Hp spaces
137(4)
4.6 The canonical factorization
141(7)
4.7 The Schwarz reflection principle for H1 functions
148(1)
4.8 Properties of outer functions
149(5)
4.9 A uniqueness theorem
154(3)
4.10 More on the norm in Hp
157(9)
Notes on
Chapter 4
163(3)
5 More function spaces
166(48)
5.1 The Nevanlinna class N
166(5)
5.2 The spectrum of b
171(2)
5.3 The disk algebra A
173(8)
5.4 The algebra C(T) + H∞
181(2)
5.5 Generalized Hardy spaces Hp(ν)
183(4)
5.6 Carleson measures
187(11)
5.7 Equivalent norms on H2
198(4)
5.8 The corona problem
202(12)
Notes on
Chapter 5
211(3)
6 Extreme and exposed points
214(43)
6.1 Extreme points
214(3)
6.2 Extreme points of LP(T)
217(2)
6.3 Extreme points of Hp
219(5)
6.4 Strict convexity
224(3)
6.5 Exposed points of B(X)
227(3)
6.6 Strongly exposed points of(B(X)
230(2)
6.7 Equivalence of rigidity and exposed points in H1
232(3)
6.8 Properties of rigid functions
235(11)
6.9 Strongly exposed points of H1
246(11)
Notes on
Chapter 6
254(3)
7 More advanced results in operator theory
257(57)
7.1 The functional calculus for self-adjoint operators
257(3)
7.2 The square root of a positive operator
260(9)
7.3 Mobius transformations and the Julia operator
269(5)
7.4 The Wold--Kolmogorov decomposition
274(1)
7.5 Partial isometries and polar decomposition
275(6)
7.6 Characterization of contractions on l2(Z)
281(1)
7.7 Densely denned operators
282(4)
7.8 Fredholm operators
286(5)
7.9 Essential spectrum of block-diagonal operators
291(7)
7.10 The dilation theory
298(8)
7.11 The abstract commutant lifting theorem
306(8)
Notes on
Chapter 7
310(4)
8 The shift operator
314(62)
8.1 The bilateral forward shift operator Zμ
314(7)
8.2 The unilateral forward shift operator S
321(7)
8.3 Commutants of Z and S
328(5)
8.4 Cyclic vectors of S
333(3)
8.5 When do we have Hp(μ) = Lp(μ)?
336(6)
8.6 The unilateral forward shift operator Sμ
342(9)
8.7 Reducing invariant subspaces of Zμ
351(2)
8.8 Simply invariant subspaces of Zμ
353(7)
8.9 Reducing invariant subspaces of Zμ
360(1)
8.10 Simply invariant subspaces of Sμ
361(2)
8.11 Cyclic vectors of Zμ and S*
363(13)
Notes on
Chapter 8
372(4)
9 Analytic reproducing kernel Hilbert spaces
376(23)
9.1 The reproducing kernel
376(5)
9.2 Multipliers
381(2)
9.3 The Banach algebra Bluit(H)
383(3)
9.4 The weak kernel
386(4)
9.5 The abstract forward shift operator S-H
390(2)
9.6 The commutant of S-H
392(2)
9.7 When do we have mult(H) = H∞?
394(2)
9.8 Invariant subspaces of SH
396(3)
Notes on
Chapter 9
396(3)
10 Bases in Banach spaces
399(55)
10.1 Minimal sequences
399(4)
10.2 Schauder basis
403(8)
10.3 The multipliers of a sequence
411(3)
10.4 Symmetric, nonsymmetric and unconditional basis
414(8)
10.5 Riesz basis
422(3)
10.6 The mappings Jx, Vx and Γx
425(5)
10.7 Characterization of the Riesz basis
430(5)
10.8 Bessel sequences and the Feichtinger conjecture
435(5)
10.9 Equivalence of Riesz and unconditional bases
440(2)
10.10 Asymptotically orthonormal sequences
442(12)
Notes on
Chapter 10
449(5)
11 Hankel operators
454(27)
11.1 A matrix representation for Hφ
454(3)
11.2 The norm of Hφ
457(5)
11.3 Hilbert's inequality
462(4)
11.4 The Nehari problem
466(4)
11.5 More approximation problems
470(3)
11.6 Finite-rank Hankel operators
473(2)
11.7 Compact Hankel operators
475(6)
Notes on
Chapter 11
478(3)
12 Toeplitz operators
481(45)
12.1 The operator Tφ ε L(H2)
481(6)
12.2 Composition of two Toeplitz operators
487(3)
12.3 The spectrum of Tφ
490(4)
12.4 The kernel of Tφ
494(5)
12.5 When is Tφ compact?
499(1)
12.6 Characterization of rigid functions
500(3)
12.7 Toeplitz operators on H2(μ)
503(3)
12.8 The Riesz projection on L2(μ)
506(5)
12.9 Characterization of invertibility
511(4)
12.10 Fredholm Toeplitz operators
515(3)
12.11 Characterization of surjectivity
518(2)
12.12 The operator X-H and its invariant subspaces
520(6)
Notes on
Chapter 12
522(4)
13 Cauchy transform and Clark measures
526(41)
13.1 The space R(D)
526(7)
13.2 Boundary behavior of Cμ
533(1)
13.3 The mapping Kμ
534(7)
13.4 The operator Kφ: L2(φ) → H2
541(4)
13.5 Functional calculus for Sφ
545(6)
13.6 Toeplitz operators with symbols in L2(T)
551(4)
13.7 Clark measures μα
555(7)
13.8 The Cauchy transform of μα
562(1)
13.9 The function ρ
563(4)
Notes on
Chapter 13
564(3)
14 Model subspaces K
567(44)
14.1 The arithmetic of inner functions
567(3)
14.2 A generator for K
570(6)
14.3 The orthogonal projection P
576(3)
14.4 The conjugation Ω
579(1)
14.5 Minimal sequences of reproducing kernels in KB
580(3)
14.6 The operators J and M
583(6)
14.7 Functional calculus for M
589(4)
14.8 Spectrum of M© and φ(M)
593(9)
14.9 The commutant lifting theorem for M
602(5)
14.10 Multipliers of K
607(4)
Notes on
Chapter 14
608(3)
15 Bases of reproducing kernels and interpolation
611(30)
15.1 Uniform minimality of (kλn)n≥1
611(1)
15.2 The Carleson--Newman condition
612(6)
15.3 Riesz basis of reproducing kernels
618(3)
15.4 Nevanlinna--Pick interpolation problem
621(2)
15.5 H∞-interpolating sequences
623(1)
15.6 H2-interpolating sequences
624(3)
15.7 Asymptotically orthonormal sequences
627(14)
Notes on
Chapter 15
638(3)
References 641(28)
Symbol index 669(4)
Author index 673(4)
Subject index 677
Emmanuel Fricain is a Professor in the Laboratoire Paul Painlevé at Université Lille 1. A part of his research focuses on the interaction between complex analysis and operator theory, which is the main matter of this book. He has a long experience of teaching numerous graduate courses on different aspects of analytic Hilbert spaces and has published several papers on H(b) spaces in high-quality journals, making him a world specialist in this subject. Javad Mashreghi is a Professor of Mathematics at Laval University, Québec, where he has been selected Star Professor of the Year five times for excellence in teaching. His main fields of interest are complex analysis, operator theory and harmonic analysis.
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