nach oben

2010 | Buch

Kapitel lesen Erstes Kapitel lesen

Harmonic Analysis of Operators on Hilbert Space

verfasst von: Béla Sz.-Nagy, Ciprian Foias, Hari Bercovici, László Kérchy

Verlag: Springer New York

Buchreihe : Universitext

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Inhaltsverzeichnis

Frontmatter

Chapter I. Contractions and Their Dilations

Abstract

In this book we study linear transformations (or “operators”) from a (real or com-plex) Hilbert space $\mathfrak{H}$ into a Hilbert space$\mathfrak{H^\prime}$; if $\mathfrak{H}=\mathfrak{H^\prime}$ we say that the transformation (or operator) is on $\mathfrak{H}.$ Note that if T is a bounded linear transformation from $\mathfrak{H}$ into $\mathfrak{H^\prime}$, then its adjoint $\mathrm{T^*}$ is the bounded linear transformation from $\mathfrak{H^\prime}$ into $\mathfrak{H}$, defined by the relation

$$\begin{array}{*{20}c}{\left( {Th,h^\prime} \right)_{\mathfrak{H}^\prime} = \left( {h,T^*h^\prime} \right)_\mathfrak{H} }& {\left( {h \in \mathfrak{H},{h^\prime} \in \mathfrak{H^\prime}} \right)} \\ \end{array}$$

we have $\|T\|=\|T^*\|$.

Béla Sz.-Nagy, Hari Bercovici, Ciprian Foias, László Kérchy

Chapter II. Geometrical and Spectral Properties of Dilations

Abstract

In the sequelwe consider a contraction T on the real or complex Hilbert space $\mathfrak{H}$, and its minimal unitary dilation U on the Hilbert space $\mathfrak{K}$, real or complex, respectively $\left( {\mathfrak{K}\supset\mathfrak{H}} \right)$. The linear manifolds

$$\begin{array}{*{20}c}{\mathfrak{L_0} = \left( {U - T} \right)\mathfrak{H}} & {{\rm and}} & {\begin{array}{*{20}c}{\mathfrak{L^*_0} = \left( {U - T^*} \right)\mathfrak{H}} \hfill & {\left( {\subset\mathfrak{K}} \right)}\hfill \\\end{array}} \hfill\\\end{array}$$

(1.1)

and their closures

$$\begin{array}{*{20}c} {\mathfrak{L} = \overline{\left( {U - T} \right)\mathfrak{H}},} \hfill & {\mathfrak{L^*} = \overline{\left( {U^* - T^*} \right)\mathfrak{H}}} \hfill \\ \end{array}$$

(1.2)

play an important role in our investigations.

Béla Sz.-Nagy, Hari Bercovici, Ciprian Foias, László Kérchy

Chapter III. Functional Calculus

Abstract

In the rest of this book we consider only complex Hilbert spaces. For the contractions T on these spaces we construct, in this chapter and in the next one, a functional calculus with the aid of the minimal unitary dilation of T. Let us begin by introducing some classes of functions, holomorphic on the open unit disk

$$\mathit{D}=\{\lambda:\:|\lambda|<1\}.$$

Béla Sz.-Nagy, Hari Bercovici, Ciprian Foias, László Kérchy

Chapter IV. Extended Functional Calculus

Abstract

We extend our functional calculus for a contraction T on $\mathfrak{K}$ so that certain unbounded functions are also allowed. Let us recall the definitions of the classes $H^\infty_T$ and $K^\infty_T$ as given in Secs. 2 and 3 of the preceding chapter: $H^\infty_T$ consists of the functions $u\in H^\infty$ ??for which the strong operator limit $u(T)=lim_{r\rightarrow1-0} u_r(T)$ exists, and $K^\infty_T$ consists of those functions $u\in H^\infty_T $ for which $u(T)^{-1}$ exists and $K^\infty_T$ is densely defined in $\mathfrak{H}$. The class $H^\infty_T$ is an algebra, and the class $K^\infty_T$ is multiplicative.

Béla Sz.-Nagy, Hari Bercovici, Ciprian Foias, László Kérchy

Chapter V. Operator-Valued Analytic Functions

Abstract

For any separable Hilbert space $\mathfrak{U}$ we denote by L²($\mathfrak{U}$) the class of functions $v(t) (0 \leq t \leq 2\pi)$ with values in $\mathfrak{U}$, measurable¹ (strongly or weakly, which are equivalent due to the separability of $\mathfrak{U}$) and such that.

$$\parallel v \parallel ^{2} = \frac{1}{2\pi} \int \limits ^{2\pi}_{0} \parallel v(t) \parallel^{2}_{\mathfrak{U}} dt < \infty.V$$

(1.1)

With this definition of the norm $\parallel v \parallel, L^{2}({\mathfrak{U}})$ becomes a (separable) Hilbert space; it is understood that two functions in $L^{2}({\mathfrak{U}})$ are considered identical if they coincide almost everywhere (with respect to Lebesgue measure). If dim ${\mathfrak{U}} = ( {\rm i.e.,} {\rm if} L^{2}({\mathfrak{U}})$ consists of scalar-valued functions), we write L ² instead of $L^{2}({\mathfrak{U}}).$

Béla Sz.-Nagy, Hari Bercovici, Ciprian Foias, László Kérchy

Chapter VI. Functional Models

Abstract

We recall the definition of the defect operators and defect spaces corresponding to a contraction T on the Hilbert space $\mathfrak{K}:$

$$\begin{array}{ll}D_{T} = (I - T ^{*} T)^{1/2},\qquad D_{T^{*}} = ({I - TT^{*}}){1/2},\\ \mathfrak{D}_{T} =\overline{D_{T}\mathfrak{H}},\qquad \qquad \quad \,\,\, \mathfrak{D}_{T^{*}} = \overline{D_{T^{*}}{\mathfrak{H}}}. \end{array}$$

Béla Sz.-Nagy, Hari Bercovici, Ciprian Foias, László Kérchy

Chapter VII. Regular Factorizations and Invariant Subspaces

Abstract

We continue our study of the geometric structure of the space of the minimal unitary (or isometric) dilation of a contraction T, given in Secs. II.1 and II.2.1. We now consider decompositions of this space induced by invariant subspaces of T. Thus, let T be a contraction on $\mathfrak{U}$ U the minimal unitary dilation of T on $\mathfrak{K}(\supset \mathfrak{H}),$ and U ₊ the minimal isometric dilation of T on $\mathfrak{K}_{+},$ where

$$\mathfrak{K}_{+} = \bigvee \limits ^{\infty}_{0} U^{n} \mathfrak{H}$$

and $U_{+} = U| \mathfrak{K}_{+}.$ We recall that, according to (I.4.2),

$$T_{*} = U^{*}_{+} | \mathfrak{H}.$$

(1.1)

Béla Sz.-Nagy, Hari Bercovici, Ciprian Foias, László Kérchy

Chapter VIII. Weak Contractions

Abstract

According to the usual definition, a self-adjoint operator A on a Hilbert space $\mathfrak{U}$, $A \geq 0$, is said to be of finite trace if A is compact (i.e., completely continuous) and the sum of its eigenvalues $\neq 0$ (each counted with the respective multiplicity) is finite. This sum is the trace of A and is denoted by tr(A).

Béla Sz.-Nagy, Hari Bercovici, Ciprian Foias, László Kérchy

Chapter IX. The Structure of C 1.-Contractions

Abstract

We systematically exploit the operators intertwining a given contraction with an isometry or unitary operator. Given operators T on $\mathfrak{N}$ and T on $\mathfrak{N^\prime}$, we denote by$\mathfrak{N^\prime}$ g $(T, T^\prime)$the set of all intertwining operators; these are the bounded linear transformations$X: \mathfrak{N} \rightarrow \mathfrak{N^\prime}$ such that $XT=T^\prime X$. We also use the notation {T} = ℐ(T,T) for the commutant of T. Fix a contraction T on $\mathfrak{H}$ an isometry (resp., unitary operator) V on ℌ, and X∈ ℐ (T, V) such that ∥X∥ ≤ 1. The pair (X,V) is called an isometric (resp., unitary) asymptote of T if for every isometry (resp., unitary operator) V′, and every X′ ∈ ℐ(T, V′) with ∥X∥ ≤ 1, there exists a unique Y ∈ ℐ(V, V′) such that V′ = Y X and ∥Y′∥ ≤ 1.

Béla Sz.-Nagy, Hari Bercovici, Ciprian Foias, László Kérchy

Chapter X. The Structure of Operators of Class C 0

Abstract

Let T be a c.n.u. contraction on the Hilbert space $\mathfrak{H}$, and$h \in \mathfrak{H}$. Denote by$\mathfrak{M_h}$ the cyclic space for T generated by h. Observe that for a function u ∈ H ^∞ we have u(T)h = 0 if and only if $u (T | \mathfrak{M_h}) = 0.$

Béla Sz.-Nagy, Hari Bercovici, Ciprian Foias, László Kérchy

Backmatter

Titel: Harmonic Analysis of Operators on Hilbert Space
verfasst von: Béla Sz.-Nagy
Ciprian Foias
Hari Bercovici
László Kérchy
Verlag: Springer New York
Electronic ISBN: 978-1-4419-6094-8
Print ISBN: 978-1-4419-6093-1
DOI: https://doi.org/10.1007/978-1-4419-6094-8

Springer Professional

Inhaltsverzeichnis

Frontmatter

Chapter I. Contractions and Their Dilations

Chapter II. Geometrical and Spectral Properties of Dilations

Chapter III. Functional Calculus

Chapter IV. Extended Functional Calculus

Chapter V. Operator-Valued Analytic Functions

Chapter VI. Functional Models

Chapter VII. Regular Factorizations and Invariant Subspaces

Chapter VIII. Weak Contractions

Chapter IX. The Structure of C 1.-Contractions

Chapter X. The Structure of Operators of Class C 0

Backmatter

Premium Partner