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 Köthe-Bochner space. An operator on $E(X)$ is called a multiplication operator if it is given by multiplication by a function in $L^{\infty}(μ).$ 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 $L^{\infty}(μ)$ 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, Rodríguez, Sánchez-Pérez in [3].

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