The boundedness condition of dilation theory characterizes subnormals and contractions
type: Item
instance of: scholarly article
title: The boundedness condition of dilation theory characterizes subnormals and contractions
zbMATH Open document ID: 0729.47019
author: Waclaw Szymanski
published in: Rocky Mountain Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: B(H) denotes the set of all linear bounded operators in a complex Hilbert space H. Let S be a semigroup. A function A: \(S\to S\to B(H)\) satisfies boundedness condition (BC) if for each \(u\in S\) there is a nonnegative real number c(u) such that \[ \sum <A(us,ut)f(s),f(t)>\leq c(u)\sum <A(s,t)f(s),f(t)>,\quad \forall f\in F(S,H), \] where F(S,H) is the linear space of all functions from S to H vanishing off a finite subset of S. The author proves that subnormality of \(T\in B(H)\) is equivalent to BC with \(A(m,n)=T^{*n}T^ m\), \(S={\mathbb{N}}.\) The analogous characterization of the contractivity of a commutative semigroup homomorphism is obtained.
Mathematics Subject Classification ID: 47B20
Mathematics Subject Classification ID: 47B44
zbMATH DE Number: 4205175
zbMATH Keywords: boundedness condition
zbMATH Keywords: subnormality
zbMATH Keywords: contractivity of a commutative semigroup homomorphism
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181073132
OpenAlex ID: W2004003330
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DOI: 10.1216/RMJM/1181073132
description: scientific article; zbMATH DE number 4205175