Multiplication operators on vector-valued function spaces (Q2845425)

This paper develops around the following general theme. Let \(V\) be a vector space of (scalar-valued or vector-valued) functions on an appropriate set and let \(A\) be an algebra of scalar-valued functions on the same set. What properties of a linear operator \(T:V\rightarrow V\) guarantee that it is an operator of multiplication by a function in \(A?\) In this respect, the authors consider a Banach function space \(E\) on a probability measure space \(\left( \Omega,\Sigma,\mu \right) \) and the Köthe-Bochner space \(E\left( X\right) \) on a given Banach space \(X\). They define a multiplication operator \(T\) on \(E\left( X\right) \) to be a map \(T:E\left( X\right) \rightarrow E\left( X\right) \) for which there exists a function \(w\in L^{\infty}\left(\mu\right) \) such that NEWLINE\[NEWLINE Tf=wf\text{ for all }f\in E\left( X\right) . NEWLINE\]NEWLINE They show that a linear operator \(T:E\left( X\right) \rightarrow E\left( X\right) \) is a multiplication operator if and only if \(T\) commutes with \(L^{\infty}\left(\mu\right) \) and leaves invariant the cyclic subspaces generated by constant functions in \(E\left( X\right) \). An alternative characterization of multiplication operators on \(E\left( X\right) \) is obtained as a consequence. Namely, a necessary and sufficient condition for an operator \(T:E\left( X\right) \rightarrow E\left( X\right) \) to be a multiplication operator is that \(T\) satisfies a functional equation introduced in [\textit{J. M. Calabuig, J. Rodríguez} and \textit{E. A. Sánchez-Pérez}, ``Multiplication operators in Köthe-Bochner spaces'', J. Math. Anal. Appl. 373, No. 1, 316--321 (2011; Zbl 1206.47026)]. For more details, the reader is encouraged to consult this interesting paper.