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Von Neumann Operators in

Published online by Cambridge University Press:  20 November 2018

Karim Seddighi
Karim Seddighi*
Affiliation:
Pennsylvania State UniversityUniversity Park, Pennsylvania16802
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Abstract

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For a connected open subset Ω of the plane and n a positive integer, let be the space introduced by Cowen and Douglas in their paper, “Complex geometry and operator theory”. Our main concern is the case n = 1, in which case we show the existence of a functional calculus for von Neumann operators in for which a spectral mapping theorem holds. In particular we prove that if the spectrum of , is a spectral set for T, and if , then σ(f(T)) = f(Ω)- for every bounded analytic function f on the interior of L, where L is compact, σ(T) ⊂ L, the interior of L is simply connected and L is minimal with respect to these properties. This functional calculus turns out to be nice in the sense that the general study of von Neumann operators in is reduced to the special situation where Ω is an open connected subset of the unit disc with .

Keywords

47B2047A6047B3747A2Von Neumann operatorfunctional calculusspectral setspectral mapping
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 2 , 01 June 1984 , pp. 146 - 156
Copyright
Copyright © Canadian Mathematical Society 1984

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