Multiplication operators in Kothe-Bochner spaces

Abstract

Let X be a Banach space and E an order continuous Banach function space over a finite measure mu. We prove that an operator T in the Kothe-Bochner space E(X) is a multiplication operator (by a function in L(infinity)(mu)) if and only if the equality T(g < f, x*> x) = g < T(f), x*> x holds for every g is an element of L(infinity)(mu), f is an element of E(X), x is an element of X and x* is an element of X*. (C) 2010 Elsevier Inc.