Boundedness criteria for Boolean algebras of projections
type: Item
instance of: scholarly article
title: Boundedness criteria for Boolean algebras of projections
zbMATH Open document ID: 0909.46017
published in: Journal of Functional Analysis
publication date: 1999-04-08T00:00:00Z
review text: For a Banach space \(X\), it is not always the case that the sum or product of two commuting spectral operators is also spectral. This is related to the Boolean algebra of projections generated by a pair of commuting bounded Boolean algebras of projections being bounded. Though results were known for \(L^p\) spaces, the author provides more general geometric conditions on \(X\) for the generated Boolean algebra of projections to be bounded. Let \(E\) and \(F\) be two commuting bounded Boolean algebras of projections on \(X\). It is shown that if \(X\) is a Banach lattice or a subspace of a \(p\)-concave Banach lattice, then the Boolean algebra generated by \(E\) and \(F\) is bounded. \(X\) is \(p\)-concave if \[ (\sum\| x_i\|^p)^{1/p}\leq M\|(\sum| x_i|^p)^{1/p}\| \] for each finite sequence in \(X\). The results are then applied to the above question of establishing if the sum or product of commuting spectral operators are spectral operators. A further application to Banach algebras of the type \(C(K)\) is considered.
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 47B40
Mathematics Subject Classification ID: 47L10
zbMATH DE Number: 1054123
zbMATH Keywords: sum or product of two commuting spectral operators
zbMATH Keywords: Boolean algebra of projections
zbMATH Keywords: \(p\)-concave Banach lattice
author: T. A. Gillespie
reviewed by: William A. Feldman
MaRDI profile type: Publication
full work available at URL: jfan.1996.3015
OpenAlex ID: W2109897541
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DOI: 10.1006/JFAN.1996.3015
description: scientific article; zbMATH DE number 1054123