Compact multipliers
Recall that a Banach \(G\)-module (resp. \(L^1(G)\)-module) is a Banach space \(E\) with an action \((x,e)\in G\times E\to L_xe\in E\) of a locally compact abelian group \(G\) (resp. \((f,e)\to T_fe\) of the convolution algebra \(L^1(G))\). In a Banach module one can define almost periodic elements as those giving rise, via the \(L^1(G)\)-action, to compact operators from \(L^1(G)\) into \(E\). An exhaustive study of Banach modules appeared in the joint work of the author with \textit{C. Datry} [Bull. Sci. Math. 119, 299-337 (1995; Zbl 0840.43005); ibid. 120, 493-536 (1996; Zbl 0871.43002)]. In the present paper multipliers from one Banach module into another (bounded linear operators which are \(L^1(G)\)-morphisms) are studied; these form a Banach module in its own right. It is shown that compact multipliers are almost periodic with the converse being valid under some natural conditions on the Banach modules in question. Some results about the approximation of compact multipliers by finite dimensional ones are also obtained.
scientific article; zbMATH DE number 1266841