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Theory of H(b) Spaces 2 Volume Hardback Set [Multiple-component retail product]

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  • Formaat: Multiple-component retail product, 1342 pages, kõrgus x laius x paksus: 237x160x94 mm, kaal: 2300 g, Worked examples or Exercises; 5 Halftones, unspecified; 25 Line drawings, unspecified, Contains 2 hardbacks
  • Sari: New Mathematical Monographs
  • Ilmumisaeg: 20-Oct-2016
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1107119413
  • ISBN-13: 9781107119413
Teised raamatud teemal:
Theory of H(b) Spaces 2 Volume Hardback SetSuurem pilt
  • Formaat: Multiple-component retail product, 1342 pages, kõrgus x laius x paksus: 237x160x94 mm, kaal: 2300 g, Worked examples or Exercises; 5 Halftones, unspecified; 25 Line drawings, unspecified, Contains 2 hardbacks
  • Sari: New Mathematical Monographs
  • Ilmumisaeg: 20-Oct-2016
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1107119413
  • ISBN-13: 9781107119413
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781107119413.html
Märksõnad:
"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"--

Arvustused

' 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 l2
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(Χ)
227(3)
6.6 Strongly exposed points of B(Χ)
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 Sμ
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 Mult(H)
383(3)
9.4 The weak kernel
386(4)
9.5 The abstract forward shift operator SH
390(2)
9.6 The commutant of SH
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 (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 subjectivity
518(2)
12.12 The operator XH 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
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
Notes on
Chapter 15
638
References 641(28)
Symbol Index 669(4)
Author Index 673(4)
Subject Index 677
Preface
16 The spaces M(A) and H(A)
16.1 The space M(A)
16.2 A characterization of M(A) ⊂ M(B)
16.3 Linear functionals on M(A)
16.4 The complementary space H(A)
16.5 The relation between H(A) and H(A*)
16.6 The overlapping space M(A) ∩ H(A)
16.7 The algebraic sum of of M(A1) and M(A2)
16.8 A decomposition of H(A)
16.9 The geometric definition of H(A)
16.10 The Julia operator J(A) and H(A)
Notes on
Chapter 16
17 Hilbert spaces inside H2
17.1 The space M(u)
17.2 The space M(u)
17.3 The space H(b)
17.4 The space H(b)
17.5 Relations between different H(b) spaces
17.6 M(u) is invariant under S and S*
17.7 Contractive inclusion of M(φ) in M(φ)
17.8 Similarity of S and SH
17.9 Invariant subspaces of Zu and Xu
17.10 An extension of Beurling's theorem
Notes on
Chapter 17
18 The structure of H(b) and H(b)
18.1 When is H(b) a closed subspace of H2?
18.2 When is H(b) a dense subset of H2?
18.3 Decomposition of H(b) spaces
18.4 The reproducing kernel of H(b)
18.5 H(b) and H(b) are invariant under Tφ
18.6 Some inhabitants of H(b)
18.7 The unilateral backward shift operators Xb and Xb
18.8 The inequality of difference quotients
18.9 A characterization of membership in H(b)
Notes on
Chapter 18
19 Geometric representation of H(b) spaces
19.1 Abstract functional embedding
19.2 A geometric representation H(b)
19.3 A unitary operator from Kb onto Kb*
19.4 A contraction from H(b) to H(b*)
19.5 Almost conformal invariance
19.6 The Littlewood Subordination Theorem revisited
19.7 The generalized Schwarz--Pick estimates
Notes on
Chapter 19
20 Representation theorems for H(b) and H(b)
20.1 Integral representation of H(b)
20.2 Kρ intertwines S*ρ and Xb
20.3 Integral representation of H(b)
20.4 A contractive antilinear map on H(b)
20.5 Absolutely continuity of the Clark measure
20.6 Inner divisors of the Cauchy transform
20.7 Vb intertwines S*μ and Xb
20.8 Analytic continuation of H(b) functions
20.9 Multipliers of H(b)
20.10 Multipliers and Toeplitz operators
20.11 Comparison of measures
Notes on
Chapter 20
21 Angular derivatives of H(b) functions
21.1 Derivative in the sense of Caratheodory
21.2 Angular derivatives and Clark measures
21.3 Derivatives of Blaschke products
21.4 Higher derivatives of b
21.5 Approximating by Blaschke products
21.6 Reproducing kernels for derivatives
21.7 An interpolation problem
21.8 Derivatives of H(b) functions
Notes on
Chapter 21
22 Bernstein-type inequalities
22.1 Passage between D and C+
22.2 Integral representations for derivatives
22.3 The weight wp,n
22.4 Some auxiliary integral operators
22.5 The operator Xp,n
22.6 Distances to the level sets
22.7 Carleson-type embedding theorems
22.8 A formula of combinatorics
22.9 Norm convergence for the reproducing kernels
Notes on
Chapter 22
23 H(b) spaces generated by a nonextreme symbol b
23.1 The pair (a, b)
23.2 Inclusion of M(u) into H(b)
23.3 The element f+
23.4 Analytic polynomials are dense in H(b)
23.5 A formula for ||Xbf||b
23.6 Another representation of H(b)
23.7 A characterization of H(b)
23.8 More inhabitants of H(b)
23.9 Unbounded Toeplitz operators and H(b) spaces
Notes on
Chapter 23
24 Operators on H(b) spaces with b nonextreme
24.1 The unilateral forward shift operator Sb
24.2 A characterization of H∞ ⊂ H(b)
24.3 Spectrum of Xb and A*b
24.4 Comparison of measures
24.5 The function Fλ
24.6 The operator Wλ
24.7 Invariant subspaces of H(b) under Xb
24.8 Completeness of the family of difference quotients
Notes on
Chapter 24
25 H(b) spaces generated by an extreme symbol b
25.1 A unitary map between H(b) and L2(ρ)
25.2 Analytic continuation of f H(b)
25.3 Analytic continuation of f H(b)
25.4 A formula for ||Xbf||b
25.5 S*-cyclic vectors in H(b) and H(b)
25.6 Orthogonal decompositions of H(b)
25.7 The closure of H(b) in H(b)
25.8 A characterization of H(b)
Notes on
Chapter 25
26 Operators on H(b) spaces with b extreme
26.1 Spectrum of Xb and X*b
26.2 Multipliers of H(b) spaces, extreme case, part I
26.3 Comparison of measures
26.4 Further characterizations of angular derivatives for b
26.5 Model operator for Hilbert space contractions
26.6 Conjugation and completeness of difference quotients
Notes on
Chapter 26
27 Inclusion between two H(b) spaces
27.1 A new geometric representation of H(b) spaces
27.2 The class I nt(Vb1, Vb2)
27.3 The class I nt(lb1, lb2)
27.4 Relations between different H(b) spaces
27.5 The rational case
27.6 Coincidence between H(b) and D(μ) spaces
Notes on
Chapter 27
28 Topics regarding inclusions M(a) ⊂ H(b) ⊂ H(b)
28.1 A sufficient and a necessary condition for H(b) = H(b)
28.2 Characterizations of H(b) = H(b)
28.3 Multipliers of H(b), extreme case, part II
28.4 Characterizations of M(a) = H(b)
28.5 Invariant subspaces of Sb when b(z) = (1 + z)/2
28.6 Characterization of bM(a) = H(b)
28.7 Characterization of the closedness of M(a) in H(b)
28.8 Boundary eigenvalues and eigenvectors of Sb*
28.9 The space H0(b)
28.10 The spectrum of S0
Notes on
Chapter 28
29 Rigid functions and strongly exposed points of H1
29.1 Admissible and special pairs
29.2 Rigid functions of H1 and H(b) spaces
29.3 Dimension of H0(b)
29.4 Sb-invariant subspaces of H(b)
29.5 A necessary condition for nonrigidity
29.6 Strongly exposed points and H(b) spaces
Notes on
Chapter 29
30 Nearly invariant subspaces and kernels of Toeplitz operators
30.1 Nearly invariant subspaces and rigid functions
30.2 The operator Rf
30.3 Extremal functions
30.4 A characterization of nearly invariant subspaces
30.5 Description of kernels of Toeplitz operators
30.6 A characterization of subjectivity for Toeplitz operators
30.7 The right inverse of a Toeplitz operator
Notes on
Chapter 30
31 Geometric properties of sequences of reproducing kernels
31.1 Completeness and minimality in H(b) spaces
31.2 Spectral properties of rank one perturbation of Xb*
31.3 Orthonormal bases in H(b) spaces
31.4 Riesz sequences of reproducing kernels in H(b)
31.5 The invertibility of distortion operator and Riesz bases
31.6 Riesz sequences in H2(μ) and in H(b)
31.7 Asymptotically orthonormal sequences and bases in H(b)
31.8 Stability of completeness and AOB
31.9 Stability of Riesz bases
Notes on
Chapter 31
References
Symbol Index
Subject Index
Emmanuel Fricain is Maître de conférences at the Institut Camille Jordan, Université Claude Bernard Lyon I. 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.