Uniqueness of translation invariant norms
type: Item
instance of: scholarly article
title: Uniqueness of translation invariant norms
zbMATH Open document ID: 0981.46006
author: Krzysztof Jarosz
published in: Journal of Functional Analysis
publication date: 2002-03-01T00:00:00Z
review text: Given a family \({\mathcal M}\) of bounded linear endomorphisms on a Banach space \((A,\|\cdot\|)\) the author searches for conditions under which \({\mathcal M}\) determines the complete norm topology of \(A\) (is the sense that any other complete norm \(|\cdot|\) on \(A\) is equivalent to \(\|\cdot\|\) provided each operator \(M\in{\mathcal M}\) is continuous in the norm \(|\cdot|\)). A special attention is paid to the case when \(A\) is a function space and the family \({\mathcal M}\) consists of mutipliers, that is operators of multiplication by a suitable function. The author proves that \({\mathcal M}\) does not determine the complete norm topology of \(A\) provided \(\text{span}(\bigcup_{M\in\mathcal M}M(A))\) has finite codimension in \(A\) and does not contain \(\bigcap_{M\in{\mathcal M}}\text{ker} M\). If \(A\) is a Banach function space on a locally compact space \(X\) and \({\mathcal M}\) is a set of multipliers on \(A\), then the above result can be reversed: \(\mathcal M\) determines the complete norm topology on \(A\) provided for any \(x\in X\) either \(\bigcap_{M\in{\mathcal M}_x}\text{ker} M=\{0\}\) or \(\text{ span}(\bigcup_{j=1}^n M_j(A))\) is finite-codimensional for some \(A_1,\dots,M_n\in{\mathcal M}_x\), where \({\mathcal M}_x=\{M-M(x):M\in{\mathcal M}\}\). Applying these results the author shows that any well-defined operator of multiplication by a non-constant function determines the complete norm topology of any Banach space \(A\) of analytic functions on an open set \(\Omega\) of \(\mathbb C^n\) separating the points of \(x\), such that all evaluation functionals are continuous on \(A\). Also the author applies the developed technique to the problem of uniqueness of a complete translation invariant norm on a translation invariant Banach space \(E\) of functions on a locally compact abelian group \(G\). He proves that the family of translations does not determine the complete norm topology on the Banach function spaces \(L^1(G)\), \(L^\inf…
reviewed by: Taras Banakh
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 46H40
Mathematics Subject Classification ID: 43A22
Mathematics Subject Classification ID: 43A25
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 43A15
Mathematics Subject Classification ID: 43A20
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 1496047
zbMATH Keywords: automatic continuity
zbMATH Keywords: uniqueness of norm
zbMATH Keywords: translation invariant functionals
MaRDI profile type: Publication
OpenAlex ID: W2091575426
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cites work: Q4072388
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DOI: 10.1006/JFAN.2000.3593
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description: scientific article; zbMATH DE number 1496047