Journal of Operator Theory
Volume 65, Issue 2, Spring 2011 pp. 235-240.
A characterization of multiplication operators on reproducing kernel Hilbert spacesAuthors: 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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