MULTIPLICATION OPERATORS ON VECTOR-VALUED FUNCTION SPACES

Abstract

Let E be a Banach function space on a probability measure space (Omega, Sigma, mu). Let X be a Banach space and E(X) be the associated Kothe-Bochner space. An operator on E(X) is called a multiplication operator if it is given by multiplication by a function in L-infinity (mu). In the main result of this paper, we show that an operator T on E(X) is a multiplication operator if and only if T commutes with L-infinity (mu) and leaves invariant the cyclic subspaces generated by the constant vector-valued functions in E(X). As a corollary we show that this is equivalent to T satisfying a functional equation considered by Calabuig, Rodriguez, and Sanchez-Perez.