Semigroups of partial isometries and symmetric operators
type: Item
instance of: scholarly article
title: Semigroups of partial isometries and symmetric operators
zbMATH Open document ID: 1226.47045
author: Robert T. W. Martin
published in: Integral Equations and Operator Theory
publication date: 2011-06-17T00:00:00Z
review text: From the author's abstract: Let \(\{ V(t)\mid t \in [0 , \infty) \}\) be a one-parameter strongly continuous semigroup of contractions on a separable Hilbert space and let \(V(-t) : = V^{*}(t)\) for \(t \in [0, \infty)\). It is shown that, if \(V(t)\) is a partial isometry for all \(t \in [-t_0 , t_0]\), \(t_0 > 0\), then the pointwise two-sided derivative of \(V(t)\) exists on a dense domain of vectors. This derivative \(B\) is necessarily a densely defined symmetric operator. This result can be viewed as a generalization of Stone's theorem for one-parameter strongly continuous unitary groups, and is used to establish sufficient conditions for a selfadjoint operator on a Hilbert space \({\mathcal{K}}\) to have a symmetric restriction to a dense linear manifold of a closed subspace \(\mathcal H \subset \mathcal K\). A large class of examples of such semigroups consisting of the compression of the unitary group generated by the operator of multiplication by the independent variable in \({\mathcal K} := \oplus_{i=1}^{n} L^2 ({\mathbb R})\) to certain model subspaces of the Hardy space of \(n\)-component vector-valued functions which are analytic in the upper half plane is presented.
reviewed by: Michael Perelmuter
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 30H10
zbMATH DE Number: 5909278
zbMATH Keywords: symmetric operators
zbMATH Keywords: Hardy space
zbMATH Keywords: model subspaces
zbMATH Keywords: strongly continuous one parameter operator semigroups
zbMATH Keywords: partial isometries
MaRDI profile type: Publication
full work available at URL: s00020-011-1878-z
OpenAlex ID: W1984192687
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DOI: 10.1007/S00020-011-1878-Z
description: scientific article; zbMATH DE number 5909278