This is an announcement for the paper "Multiplication operators on vector-valued function spaces" by Hulya Duru, Arkady Kitover, and Mehmet Orhon.
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].
Archive classification: math.FA
Mathematics Subject Classification: 47B38 (Primary) 46G10, 46B42, 46H25 (Secondary)
Submitted from: mo@unh.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1104.2806
or
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alspach＠math.okstate.edu