Multiplication operators on vector-valued function spaces

TL;DR

This paper characterizes multiplication operators on vector-valued function spaces as those commuting with scalar multiplication and preserving certain cyclic subspaces, linking operator properties to functional equations.

Contribution

It provides a new characterization of multiplication operators on vector-valued function spaces through commutation and invariance properties.

Findings

Operator T is a multiplication operator iff it commutes with L^{}() and preserves cyclic subspaces.

The characterization links operator properties to a specific functional equation.

Results extend understanding of operator structure on Banach and Kf6the-Bochner 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 K\"{o}the-Bochner space. An operator on $E(X)$ is called a multiplication operator if it is given by multiplication by a function in $L^{\infty}(\mu).$ 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 $L^{\infty}(\mu)$ 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, Rodr\'{i}guez, S\'{a}nchez-P\'{e}rez in [3].