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# Finite rank operators and functional calculus on Hilbert modules over abelian $C^{\ast}$-algebras

Published:1997-06-01
Printed: Jun 1997
• Dan Kucerovsky
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## Abstract

We consider the problem: If $K$ is a compact normal operator on a Hilbert module $E$, and $f\in C_0(\Sp K)$ is a function which is zero in a neighbourhood of the origin, is $f(K)$ of finite rank? We show that this is the case if the underlying $C^{\ast}$-algebra is abelian, and that the range of $f(K)$ is contained in a finitely generated projective submodule of $E$.
 MSC Classifications: 55R50 - Stable classes of vector space bundles, $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19-XX} 47A60 - Functional calculus 47B38 - Operators on function spaces (general)