Compact weighted composition operators on Banach lattices

Abstract

Abstract.A characterization of compact (and M-weakly compact) weighted composition operators on real and complex Banach lattices which can be appropriately realized as function spaces is provided.The compact weighted composition operators on C (X), all bounded continuous complex valued functions on X with the supremum norm, have been characterized in the case of compact X by Kamowitz [4], and for completely regular X by Singh and Summers [6]. We recall that an operator F on C (X) is called a weighted composition operator if there exists an r on C (X) and a continuous map <f> from X to X so that for each / in C (X), we haveIn this note we will characterize a class of compact weighted composition operators (and compact composition operators) on more general function spaces, which we will call Banach F-lattices. These include the Lp -spaces and the Banach lattices, both real and complex, with quasi-interior points (more generally, topological order partitions). We also demonstrate that these operators are equivalent to the A/-weakly compact weighted composition operators.We will call a complex Banach space E a Banach F-lattice if there exists a real Banach lattice G, which can be identified with equivalence classes of real valued functions on a completely regular space X, and E = G + iG can be identified with equivalence classes of complex valued functions on X satisfying the following conditions:(i) If / is equivalent to g , then {x. G X: f(x) = g(x)} is dense.(ii) Each continuous complex valued bounded function on X (or function with compact support in the case of a locally compact X ) represents an element in E. (iii) The vector space operations on E and the lattice operations on G correspond to the pointwise defined operations.

A survey of weighted substitution operators and generalizations of Banach‐stone theorem

“…Theory of these operators had extensive developments during the last four decades or so and they have been studied on several function spaces of holomorphic functions, continuous functions, and measurable functions and their combinations. For details we refer to [8,10,11,15,25,28].…”