Cover

Half-title

Series information

Title page

Copyright information

Dedication

Contents for Volume 2

Contents for Volume 1

Preface

16 The spaces mathcal M(A) and mathcal H(A)

16.1 The space mathcal M(A)

16.2 A characterization of mathcal M(A) subset mathcal M(B)

16.3 Linear functionals on mathcal M(A)

16.4 The complementary space mathcal H(A)

16.5 The relation between mathcal H(A) and mathcal H(A*)

16.6 The overlapping space mathcal M(A) cap mathcal H(A)

16.7 The algebraic sum of mathcal M(A[sub(1)]) and mathcal M(A[sub(2)])

16.8 A decomposition of mathcal H(A)

16.9 The geometric definition of mathcal H(A)

16.10 The Julia operator mathfrak J(A) and mathcal H(A)

Notes on Chapter 16

17 Hilbert spaces inside H[sup(2)]

17.1 The space mathcal M(u)

17.2 The space mathcal M(bar u)

17.3 The space mathcal H(b)

17.4 The space mathcal H(bar b)

17.5 Relations between different mathcal H(bar b) spaces

17.6 mathcal M(bar u) is invariant under S and S*

17.7 Contractive inclusion of mathcal M(u) in mathcal M(bar u)

17.8 Similarity of S and S[sub(mathcal H)]

17.9 Invariant subspaces of Z[sub(bar u)] and X[sub(bar u)]

17.10 An extension of Beurling's theorem

Notes on Chapter 17

18 The structure of mathcal H(b) and mathcal H(bar b)

18.1 When is mathcal H(b) a closed subspace of H[sup(2)]?

18.2 When is mathcal H(b) a dense subset of H[sup(2)]?

18.3 Decomposition of mathcal H(b) spaces

18.4 The reproducing kernel of mathcal H(b)

18.5 mathcal H(b) and mathcal H(bar b) are invariant under T[sub(bar varphi)]

18.6 Some inhabitants of mathcal H(b)

18.7 The unilateral backward shift operators X[sub(b)] and X[sub(bar b)]

18.8 The inequality of difference quotients

18.9 A characterization of membership in mathcal H(b)

Notes on Chapter 18

19 Geometric representation of mathcal H(b) spaces

19.1 Abstract functional embedding

19.2 A geometric representation of mathcal H(b)

19.3 A unitary operator from mathbb K[sub(b)] onto mathbb K[sub(b*)]

19.4 A contraction from mathcal H(b) to mathcal 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 mathcal H(b) and mathcal H(bar b)

20.1 Integral representation of mathcal H(bar b)

20.2 K[sub(ρ)] intertwines S[sub(ρ)]* and X[sub(bar b)]

20.3 Integral representation of mathcal H(b)

20.4 A contractive antilinear map on mathcal H(b)

20.5 Absolute continuity of the Clark measure

20.6 Inner divisors of the Cauchy transform

20.7 V[sub(b)] intertwines S[sub(μ)]* and X[sub(b)]

20.8 Analytic continuation of mathcal H(b) functions

20.9 Multipliers of mathcal H(b)

20.10 Multipliers and Toeplitz operators

20.11 Comparison of measures

Notes on Chapter 20

21 Angular derivatives of mathcal H(b) functions

21.1 Derivative in the sense of Carathéodory

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 mathcal H(b) functions

Notes on Chapter 21

22 Bernstein-type inequalities

22.1 Passage between mathbb D and mathbb C[sub(+)]

22.2 Integral representations for derivatives

22.3 The weight w[sub(p,n)]

22.4 Some auxiliary integral operators

22.5 The operator T[sub(p,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 mathcal H(b) spaces generated by a nonextreme symbol b

23.1 The pair (a,b)

23.2 Inclusion of mathcal M(u) into mathcal H(b)

23.3 The element f[sup(+)]

23.4 Analytic polynomials are dense in mathcal H(b)

23.5 A formula for ||X[sub(b)]f||[sub(b)]

23.6 Another representation of mathcal H(b)

23.7 A characterization of mathcal H(b)

23.8 More inhabitants of mathcal H(b)

23.9 Unbounded Toeplitz operators and mathcal H(b) spaces

Notes on Chapter 23

24 Operators on mathcal H(b) spaces with b nonextreme

24.1 The unilateral forward shift operator S[sub(b)]

24.2 A characterization of H[sup(∞)] subset mathcal H(b)

24.3 Spectrum of X[sub(b)] and X[sub(b)]*

24.4 Comparison of measures

24.5 The function F[sub(λ)]

24.6 The operator W[sub(λ)]

24.7 Invariant subspaces of mathcal H(b) under X[sub(b)]

24.8 Completeness of the family of difference quotients

Notes on Chapter 24

25 mathcal H(b) spaces generated by an extreme symbol b

25.1 A unitary map between mathcal H(bar b) and L[sup(2)](ρ)

25.2 Analytic continuation of f in mathcal H(bar b)

25.3 Analytic continuation of f in mathcal H(b)

25.4 A formula for ||X[sub(b)] f||[sub(b)]

25.5 S*-cyclic vectors in mathcal H(b) and mathcal H(bar b)

25.6 Orthogonal decompositions of mathcal H(b)

25.7 The closure of mathcal H(bar b) in mathcal H(b)

25.8 A characterization of mathcal H(b)

Notes on Chapter 25

26 Operators on mathcal H(b) spaces with b extreme

26.1 Spectrum of X[sub(b)] and X[sub(b)]*

26.2 Multipliers of mathcal 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 mathcal H(b) spaces

27.1 A new geometric representation of mathcal H(b) spaces

27.2 The class mathscr I nt(V[sub(b1)],V[sub(b2)])

27.3 The class mathscr I nt(maths S[sub(b1)], mathscr S[sub(b2)])

27.4 Relations between different mathcal H(b) spaces

27.5 The rational case

27.6 Coincidence between mathcal H(b) and mathcal D(μ) spaces

Notes on Chapter 27

28 Topics regarding inclusions mathcal M(a) subset mathcal H(bar b) subset mathcal H(b)

28.1 A necessary and sufficient condition for mathcal H(bar b) = mathcal H(b)

28.2 Characterizations of mathcal H(bar b) = mathcal H(b)

28.3 Multipliers of mathcal H(b) spaces, extreme case, part II

28.4 Characterizations of mathcal M(a) = mathcal H(b)

28.5 Invariant subspaces of S[sub(b)] when b(z)=(1+z)/2

28.6 Characterization of overline mathcal M(a)[sup(b)] = mathcal H(b)

28.7 Characterization of the closedness of mathcal M(a) in mathcal H(b)

28.8 Boundary eigenvalues and eigenvectors of S[sub(b)]*

28.9 The space mathcal H[sub(0)](b)

28.10 The spectrum of S[sub(0)]

Notes on Chapter 28

29 Rigid functions and strongly exposed points of H[sup(1)]

29.1 Admissible and special pairs

29.2 Rigid functions of H[sup(1)] and mathcal H(b) spaces

29.3 Dimension of mathcal H[sub(0)](b)

29.4 S[sub(b)]-invariant subspaces of mathcal H(b)

29.5 A necessary condition for nonrigidity

29.6 Strongly exposed points and mathcal 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 R[sub(f)]

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 surjectivity 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 mathcal H(b) spaces

31.2 Spectral properties of rank-one perturbation of X[sub(b)]*

31.3 Orthonormal bases in mathcal H(b) spaces

31.4 Riesz sequences of reproducing kernels in mathcal H(b)

31.5 The invertibility of distortion operator and Riesz bases

31.6 Riesz sequences in H[sup(2)](μ) and in mathcal H(bar b)

31.7 Asymptotically orthonormal sequences and bases in mathcal H(b)

31.8 Stability of completeness and asymptotically orthonormal basis

31.9 Stability of Riesz bases

Notes on Chapter 31

References

Symbol index

Author index

Subject index