Uniqueness of translation invariant norms
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 \(L^1(G)\), \(L^\infty(G)\) or \(C(G)\) on a compact abelian group \(G\). On the other hand, the family of all rotations of the circle \(\mathbb T\) determines the complete norm topology of \(L^p(\mathbb T)\) for \(1<p<\infty\), while any finite family of rotations does not. In contrast, any single non-identity translation determines the complete norm topology of \(L^1(\mathbb R)\).
scientific article; zbMATH DE number 1496047
scientific article