MULTIPLICATION OPERATORS ON VECTOR-VALUED FUNCTION SPACES.

Let E be a Banach function space on a probability measure space (Ω,Σ,μ). 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∞(μ). 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∞(μ) 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, Rodr'ıguez, and S'anchez-P'erez. [ABSTRACT FROM AUTHOR]