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# Journal of Operator Theory

Volume 65, Issue 2, Spring 2011  pp. 235-240.

A characterization of multiplication operators on reproducing kernel Hilbert spaces

Authors Christoph Barbian
Author institution: Fachrichtung 6.1 (Mathematik), Universitaet des Saarlandes, D-66041 Saarbruecken, Germany

Summary:  In this note, we prove that an operator between reproducing kernel Hilbert spaces is a multiplication operator if and only if it leaves invariant zero sets. To be more precise, it is shown that an operator $T$ between reproducing kernel Hilbert spaces is a multiplication operator if and only if $(Tf)(z) = 0$ holds for all $f$ and $z$ satisfying $f(z) = 0$. As possible applications, we deduce a general reflexivity result for multiplier algebras, and furthermore prove fully vector-valued generalizations of multiplier lifting results of Beatrous and Burbea.

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