On quotients and restrictions of generalized scalar operators
type: Item
instance of: scholarly article
title: On quotients and restrictions of generalized scalar operators
zbMATH Open document ID: 0678.47029
DOI: 10.1016/0022-1236(89)90113-4
author: Jörg Eschmeier
author: Mihai Putinar
published in: Journal of Functional Analysis
publication date: 1989-01-01T00:00:00Z
review text: The authors show that an operator \(T\in L(X)\) on a Banach space X is a quotient of a generalized scalar operator if and only if \[ (*)\quad T_ z: {\mathcal E}'({\mathbb{C}}){\tilde \otimes}X\to {\mathcal E}'({\mathbb{C}}){\tilde \otimes}X,\quad u\mapsto (z-T)u \] is surjective. If \({\mathbb{C}}\) is replaced by the real numbers \({\mathbb{R}}^ a \)spectral characterization of those T is given obeying this modified condition (*). This characterization is used to solve division problems for vector valued distributions on \({\mathbb{R}}\). Finally, an example is exhibited of an operator, which is both a restriction and a quotient of a generalized scalar operator, but which is not even strongly decomposable. This is in strict contrast to the situation with decomposable operators.
Mathematics Subject Classification ID: 47B40
Mathematics Subject Classification ID: 46F10
zbMATH DE Number: 4111384
zbMATH Keywords: quotient of a generalized scalar operator
zbMATH Keywords: spectral characterization
zbMATH Keywords: division problems for vector valued distributions
zbMATH Keywords: an operator, which is both a restriction and a quotient of a generalized scalar operator, but which is not even strongly decomposable
reviewed by: Volker Wrobel
MaRDI profile type: Publication
full work available at URL: 0022-1236(89)90113-4
OpenAlex ID: W1984908324
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description: scientific article; zbMATH DE number 4111384