Multiplication operators on vector-valued function spaces

Details Multiplication operators on vector-valued function spaces Multiplication operators on vector-valued function spaces

2011-04-14

arxiv.org/abs/1104.2806v1

Mathematics

Functional Analysis

Scientific paper

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].

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