Multiplication operators on vector-valued function spaces

Abstract

Let E E be a Banach function space on a probability measure space ( Ω , Σ , μ ) . (\Omega ,\Sigma ,\mu ). Let X X be a Banach space and E ( X ) E(X) be the associated Köthe-Bochner space. An operator on E ( X ) E(X) is called a multiplication operator if it is given by multiplication by a function in L ∞ ( μ ) . L^{\infty }(\mu ). In the main result of this paper, we show that an operator T T on E ( X ) E(X) is a multiplication operator if and only if T T commutes with L ∞ ( μ ) L^{\infty }(\mu ) and leaves invariant the cyclic subspaces generated by the constant vector-valued functions in E ( X ) . E(X). As a corollary we show that this is equivalent to T T satisfying a functional equation considered by Calabuig, Rodríguez, and Sánchez-Pérez.