Chapter
1.10 Invariant subspaces and cyclic vectors
1.11 Compressions and dilations
1.12 Angle between two subspaces
2 Some families of operators
2.1 Finite-rank operators
2.3 Subdivisions of spectrum
2.4 Self-adjoint operators
2.6 Normal and unitary operators
2.7 Forward and backward shift operators on ell[sup(2)]
2.8 The multiplication operator on L[sup(2)](μ)
2.9 Doubly infinite Toeplitz and Hankel matrices
3 Harmonic functions on the open unit disk
3.1 Nontangential boundary values
3.3 Some well-known facts in measure theory
3.4 Boundary behavior of P(μ)
3.5 Integral means of P(μ)
3.6 Boundary behavior of Q(μ)
3.7 Integral means of Q(μ)
3.8 Subharmonic functions
3.9 Some applications of Green’s formula
4.2 Classic Hardy spaces H[sup(p)]
4.3 The Riesz projection P[sub(+)]
4.4 Kernels of P[sub(+)] and P[sub(-)]
4.5 Dual and predual of H[sup(p)]
spaces
4.6 The canonical factorization
4.7 The Schwarz reflection principle for H[sup(1)]
functions
4.8 Properties of outer functions
4.10 More on the norm in H[sup(p)]
5.1 The Nevanlinna class mathcal N
5.3 The disk algebra
mathcal A
5.4 The algebra mathcal C( mathbb T)+H[sup(∞)]
5.5 Generalized Hardy spaces H[sup(p)](nu)
5.7 Equivalent norms on H[sup(2)]
6 Extreme and exposed points
6.2 Extreme points of L[sup(p)](mathbb T)
6.3 Extreme points of H[sup(p)]
6.5 Exposed points of mathfrak B( mathcal X)
6.6 Strongly exposed points of
mathfrak B( mathcal X)
6.7 Equivalence of rigidity and exposed points in H[sup(1)]
6.8 Properties of rigid functions
6.9 Strongly exposed points of H[sup(1)]
7 More advanced results in operator theory
7.1 The functional calculus for self-adjoint operators
7.2 The square root of a positive operator
7.3 Möbius transformations and the Julia operator
7.4 The Wold–Kolmogorov decomposition
7.5 Partial isometries and polar decomposition
7.6 Characterization of contractions on ell[sup(2)](mathbb Z)
7.7 Densely defined operators
7.9 Essential spectrum of block-diagonal operators
7.11 The abstract commutant lifting theorem
8.1 The bilateral forward shift operator Z[sub(μ)]
8.2 The unilateral forward shift operator S
8.3 Commutants of Z and S
8.5 When do we have H[sup(p)](μ) = L[sup(p)](μ)?
8.6 The unilateral forward shift operator S[sub(μ)]
8.7 Reducing invariant subspaces of Z[sub(μ)]
8.8 Simply invariant subspaces of Z[sub(μ)]
8.9 Reducing invariant subspaces of S[sub(μ)]
8.10 Simply invariant subspaces of S[sub(μ)]
8.11 Cyclic vectors of Z[sub(μ)] and S[sup(*)]
9 Analytic reproducing kernel Hilbert spaces
9.1 The reproducing kernel
9.3 The Banach algebra mathfrak Mult(mathcal H)
9.5 The abstract forward shift operator S[sub(mathcal H)]
9.6 The commutant of S[sub(mathcal H)]
9.7 When do we have [mathfrac Mult( mathcal H)]=H[sup(∞)]?
9.8 Invariant subspaces of S[sub(mathcal H)]
10 Bases in Banach spaces
10.3 The multipliers of a sequence
10.4 Symmetric, nonsymmetric and unconditional basis
10.6 The mappings J[sub(mathfrak X)], V[sub(mathfrac X)] and Γ[sub(mathfrac X)]
10.7 Characterization of the Riesz basis
10.8 Bessel sequences and the Feichtinger conjecture
10.9 Equivalence of Riesz and unconditional bases
10.10 Asymptotically orthonormal sequences
11.1 A matrix representation for H sub(varphi)
11.2 The norm of H[sub(varphi)]
11.3 Hilbert’s inequality
11.5 More approximation problems
11.6 Finite-rank Hankel operators
11.7 Compact Hankel operators
12.1 The operator T[sub(varphi)] in mathcal L(H[sub(2)])
12.2 Composition of two Toeplitz operators
12.3 The spectrum of T[sub(varphi)]
12.4 The kernel of T[sub(varphi])
12.5 When is T[sub(varphi)] compact?
12.6 Characterization of rigid functions
12.7 Toeplitz operators on H[sup(2)](μ)
12.8 The Riesz projection on L[sup(2)](μ)
12.9 Characterization of invertibility
12.10 Fredholm Toeplitz operators
12.11 Characterization of surjectivity
12.12 The operator X[sup(mathcal H)] and its invariant subspaces
13 Cauchy transform and Clark measures
13.1 The space mathfrak K(mathbb D)
13.2 Boundary behavior of C[sub(μ)]
13.3 The mapping K[sub(μ)]
13.4 The operator K[sub(varphi)]: L[sup(2)](varphi) →H[sup(2)]
13.5 Functional calculus for S[sub(varphi)]
13.6 Toeplitz operators with symbols in L[sup(2)](mathbb T)
13.7 Clark measures μ[sub(α)]
13.8 The Cauchy transform of μ[sub(α)]
14 Model subspaces K[sub(Θ)]
14.1 The arithmetic of inner functions
14.2 A generator for K[sub(Θ)]
14.3 The orthogonal projection P[sub(Θ)]
14.4 The conjugation Ω[sub(Θ)]
14.5 Minimal sequences of reproducing kernels in K[sub(B)]
14.6 The operators J and M[sub(Θ)]
14.7 Functional calculus for M[sub(Θ)]
14.8 Spectrum of M[sub(Θ)] and varphi(M[sub(Θ)])
14.9 The commutant lifting theorem for M[sub(Θ)]
14.10 Multipliers of K[sub(Θ)]
15 Bases of reproducing kernels and interpolation
15.1 Uniform minimality of (k[sub(λn)])[sub(n≥1)]
15.2 The Carleson–Newman condition
15.3 Riesz basis of reproducing kernels
15.4 Nevanlinna–Pick interpolation problem
15.5 H[sup(∞)]-interpolating sequences
15.6 H[sup(2)]-interpolating sequences
15.7 Asymptotically orthonormal sequences
The Theory of H(b) Spaces: Volume 1
( New Mathematical Monographs )
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