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E-raamat: Theory of H(b) Spaces: Volume 2

(Université Lyon I), (Université Laval, Québec)
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  • Sari: New Mathematical Monographs
  • Ilmumisaeg: 20-Oct-2016
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781316348925
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Theory of H(b) Spaces: Volume 2Suurem pilt
  • Formaat: EPUB+DRM
  • Sari: New Mathematical Monographs
  • Ilmumisaeg: 20-Oct-2016
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781316348925
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An H(b) space is defined as a collection of analytic functions that 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. Volume 1 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. Volume 2 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.

Arvustused

'The monograph contains numerous references to original papers, as well as numerous exercises. This monograph may be strongly recommended as a good introduction to this interesting and intensively developing branch of analysis ' Vladimir S. Pilidi, Zentralblatt MATH 'As with Volume 1, chapter notes outline historical development, and an extensive bibliography cites substantial work done in the area since 2000.' Joseph D. Lakey, MathSciNet ' 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
16 The spaces M(A) and H(A)
1(43)
16.1 The space M(A)
1(7)
16.2 A characterization of M(A) ⊂ M(B)
8(6)
16.3 Linear functionals on M(A)
14(1)
16.4 The complementary space H(A)
15(3)
16.5 The relation between H(A) and H(A*)
18(2)
16.6 The overlapping space M(A) ∩ H(A)
20(1)
16.7 The algebraic sum of M(A1) and M(A2)
21(4)
16.8 A decomposition of H(A)
25(7)
16.9 The geometric definition of H(A)
32(7)
16.10 The Julia operator J(A) and H(A)
39(5)
Notes on
Chapter 16
41(3)
17 Hilbert spaces inside H2
44(25)
17.1 The space M(u)
44(3)
17.2 The space M(u)
47(2)
17.3 The space H(b)
49(2)
17.4 The space H(b)
51(2)
17.5 Relations between different H(b) spaces
53(2)
17.6 M(u) is invariant under S and S*
55(1)
17.7 Contractive inclusion of M(u) in M(u)
56(2)
17.8 Similarity of S and SH
58(4)
17.9 Invariant subspaces of Zu and Xu
62(2)
17.10 An extension of Beurling's theorem
64(5)
Notes on
Chapter 17
67(2)
18 The structure of H(b) and H(b)
69(30)
18.1 When is H(b) a closed subspace of H2?
69(2)
18.2 When is H(b) a dense subset of H2?
71(2)
18.3 Decomposition of H(b) spaces
73(2)
18.4 The reproducing kernel of H(b)
75(2)
18.5 H(b) and H(b) are invariant under Tφ
77(3)
18.6 Some inhabitants of H(b)
80(5)
18.7 The unilateral backward shift operators Xb and Xb
85(5)
18.8 The inequality of difference quotients
90(1)
18.9 A characterization of membership in H(b)
91(8)
Notes on
Chapter 18
96(3)
19 Geometric representation of H(b) spaces
99(35)
19.1 Abstract functional embedding
99(9)
19.2 A geometric representation of H(b)
108(4)
19.3 A unitary operator from Kb onto Kb
112(5)
19.4 A contraction from H(b) to H(b*)
117(7)
19.5 Almost conformal invariance
124(3)
19.6 The Littlewood subordination theorem revisited
127(2)
19.7 The generalized Schwarz--Pick estimates
129(5)
Notes on
Chapter 19
131(3)
20 Representation theorems for H(b) and H(b)
134(36)
20.1 Integral representation of H(b)
135(3)
20.2 Kρ intertwines S*ρ and Xb
138(1)
20.3 Integral representation of H(b)
139(5)
20.4 A contractive antilinear map on H(b)
144(2)
20.5 Absolute continuity of the Clark measure
146(1)
20.6 Inner divisors of the Cauchy transform
147(2)
20.7 Vb intertwines S*μ and Xb
149(2)
20.8 Analytic continuation of H(b) functions
151(3)
20.9 Multipliers of H(b)
154(2)
20.10 Multipliers and Toeplitz operators
156(6)
20.11 Comparison of measures
162(8)
Notes on
Chapter 20
167(3)
21 Angular derivatives of H(b) functions
170(54)
21.1 Derivative in the sense of Caratheodory
171(11)
21.2 Angular derivatives and Clark measures
182(4)
21.3 Derivatives of Blaschke products
186(5)
21.4 Higher derivatives of b
191(5)
21.5 Approximating by Blaschke products
196(8)
21.6 Reproducing kernels for derivatives
204(3)
21.7 An interpolation problem
207(7)
21.8 Derivatives of H(b) functions
214(10)
Notes on
Chapter 21
220(4)
22 Bernstein-type inequalities
224(49)
22.1 Passage between D and C+
225(3)
22.2 Integral representations for derivatives
228(8)
22.3 The weight wp,n
236(2)
22.4 Some auxiliary integral operators
238(10)
22.5 The operator Tp,n
248(3)
22.6 Distances to the level sets
251(5)
22.7 Carleson-type embedding theorems
256(6)
22.8 A formula of combinatorics
262(4)
22.9 Norm convergence for the reproducing kernels
266(7)
Notes on
Chapter 22
271(2)
23 H(b) spaces generated by a nonextreme symbol b
273(42)
23.1 The pair (a, b)
274(3)
23.2 Inclusion of M(u) into H(b)
277(1)
23.3 The element f+
278(6)
23.4 Analytic polynomials are dense in H(b)
284(1)
23.5 A formula for ||Xbf||b
285(4)
23.6 Another representation of H(b)
289(4)
23.7 A characterization of H(b)
293(12)
23.8 More inhabitants of H(b)
305(3)
23.9 Unbounded Toeplitz operators and H(b) spaces
308(7)
Notes on
Chapter 23
312(3)
24 Operators on H(b) spaces with b nonextreme
315(38)
24.1 The unilateral forward shift operator Sb
316(6)
24.2 A characterization of H∞ ⊂ H(b)
322(5)
24.3 Spectrum of Xb and X*b
327(2)
24.4 Comparison of measures
329(3)
24.5 The function Fλ
332(3)
24.6 The operator Wλ
335(10)
24.7 Invariant subspaces of H(b) under Xb
345(3)
24.8 Completeness of the family of difference quotients
348(5)
Notes on
Chapter 24
350(3)
25 H(b) spaces generated by an extreme symbol b
353(24)
25.1 A unitary map between H(b) and L2(ρ)
354(2)
25.2 Analytic continuation of f e H(b)
356(1)
25.3 Analytic continuation of f e H(b)
357(3)
25.4 A formula for ||Xbf||b
360(2)
25.5 S*-cyclic vectors in H(b) and H(b)
362(2)
25.6 Orthogonal decompositions of H(b)
364(1)
25.7 The closure of H(b) in H(b)
365(2)
25.8 A characterization of H(b)
367(10)
Notes on
Chapter 25
375(2)
26 Operators on H(b) spaces with b extreme
377(25)
26.1 Spectrum of Xb and X*b
378(3)
26.2 Multipliers of H(b) spaces, extreme case, part I
381(3)
26.3 Comparison of measures
384(4)
26.4 Further characterizations of angular derivatives for b
388(3)
26.5 Model operator for Hilbert space contractions
391(3)
26.6 Conjugation and completeness of difference quotients
394(8)
Notes on
Chapter 26
399(3)
27 Inclusion between two H(b) spaces
402(37)
27.1 A new geometric representation of H(b) spaces
403(4)
27.2 The class Int(Vb1, Vb2)
407(5)
27.3 The class Int(lb1, lb2)
412(5)
27.4 Relations between different H(b) spaces
417(8)
27.5 The rational case
425(7)
27.6 Coincidence between H(b) and D(μ) spaces
432(7)
Notes on
Chapter 27
436(3)
28 Topics regarding inclusions M(a) ⊂ H(b) ⊂ H(b)
439(50)
28.1 A necessary and sufficient condition for H(b) = H(b)
440(3)
28.2 Characterizations of H(b) = H(b)
443(8)
28.3 Multipliers of H(b) spaces, extreme case, part II
451(6)
28.4 Characterizations of M(a) = H(b)
457(7)
28.5 Invariant subspaces of Sb when b(z) = (1 + z)/2
464(11)
28.6 Characterization of M(a)b = H(b)
475(1)
28.7 Characterization of the closedness of M(a) in H(b)
476(1)
28.8 Boundary eigenvalues and eigenvectors of S*b
477(5)
28.9 The space H0(b)
482(3)
28.10 The spectrum of S0
485(4)
Notes on
Chapter 28
486(3)
29 Rigid functions and strongly exposed points of H1
489(31)
29.1 Admissible and special pairs
489(3)
29.2 Rigid functions of H1 and H(b) spaces
492(5)
29.3 Dimension of H0(b)
497(7)
29.4 Sb-invariant subspaces of H(b)
504(3)
29.5 A necessary condition for nonrigidity
507(4)
29.6 Strongly exposed points and H(b) spaces
511(9)
Notes on
Chapter 29
518(2)
30 Nearly invariant subspaces and kernels of Toeplitz operators
520(33)
30.1 Nearly invariant subspaces and rigid functions
520(2)
30.2 The operator Rf
522(4)
30.3 Extremal functions
526(3)
30.4 A characterization of nearly invariant subspaces
529(7)
30.5 Description of kernels of Toeplitz operators
536(7)
30.6 A characterization of surjectivity for Toeplitz operators
543(3)
30.7 The right-inverse of a Toeplitz operator
546(7)
Notes on
Chapter 30
551(2)
31 Geometric properties of sequences of reproducing kernels
553(50)
31.1 Completeness and minimality in H(b) spaces
554(7)
31.2 Spectral properties of rank-one perturbation of X*b
561(3)
31.3 Orthonormal bases in H(b) spaces
564(3)
31.4 Riesz sequences of reproducing kernels in H(b)
567(4)
31.5 The invertibility of distortion operator and Riesz bases
571(13)
31.6 Riesz sequences in H2(μ) and in H(b)
584(1)
31.7 Asymptotically orthonormal sequences and bases in H(b)
585(3)
31.8 Stability of completeness and asymptotically orthonormal basis
588(7)
31.9 Stability of Riesz bases
595(8)
Notes on
Chapter 31
600(3)
References 603(11)
Symbol Index 614(2)
Author Index 616(2)
Subject Index 618
Emmanuel Fricain is Professor of Mathematics at Laboratoire Paul Painlevé, Université Lille 1, France. Part of his research focuses on the interaction between complex analysis and operator theory, which is the main content of this book. He has a wealth of experience teaching numerous graduate courses on different aspects of analytic Hilbert spaces, and he 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 the Université Laval, Québec, Canada, where he has been selected Star Professor of the Year seven times for excellence in teaching. His main fields of interest are complex analysis, operator theory and harmonic analysis. He is the author of several mathematical textbooks, monographs and research articles. He won the G. de B. Robinson Award, the publication prize of the Canadian Mathematical Society, in 2004.
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