Mean\(_2\)-bounded operators on Hilbert space and weight sequences of positive operators
type: Item
instance of: scholarly article
title: Mean\(_2\)-bounded operators on Hilbert space and weight sequences of positive operators
zbMATH Open document ID: 0946.47020
DOI: 10.1023/A:1009794510984
author: Earl Berkson
author: T. A. Gillespie
published in: Positivity
publication date: 2000-01-11T00:00:00Z
review text: Let \(H\) be a complex Hilbert space. An operator-valued weight sequence on \(H\) is a bilateral sequence \(W= \{W_k\}^\infty_{k= -\infty}\) whose terms are bounded, positive, invertible self-adjoint operators. The authors introduce and study the class of \(\text{mean}_2\)-bounded operators on Hilbert space. These operators, which are characterized by a uniform boundedness condition on the ``quadratic averages'' of their powers, are intimately related to operator-valued weight sequences and weighted norm inequalities. Under certain circumstances, \(\text{mean}_2\)-bounded operators have a rich spectral theory. Much of the paper is taken up with a discussion of relationships between the classes of trigonometrically well-bounded operators and the \(\text{mean}_2\)-bounded operators. The former class was introduced by the authors in AC functions on the circle and spectral families [J. Oper. Theory 13, 33-47 (1985; Zbl 0566.47011)]. The present paper develops further ideas in the papers by \textit{S. Treil} and \textit{A. Volberg} [J. Funct. Analysis 143, 269-308 (1997; Zbl 0876.42027)].
Mathematics Subject Classification ID: 47B40
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 42A45
zbMATH DE Number: 1339335
zbMATH Keywords: quadratic averages
zbMATH Keywords: operator-valued weight sequence
zbMATH Keywords: invertible self-adjoint operators
zbMATH Keywords: \(\text{mean}_2\)-bounded operators
zbMATH Keywords: uniform boundedness
zbMATH Keywords: trigonometrically well-bounded operators
zbMATH Keywords: AC functions
reviewed by: H. R. Dowson
MaRDI profile type: Publication
full work available at URL: a:1009794510984
OpenAlex ID: W60062045
description: scientific article; zbMATH DE number 1339335