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Similarity Classification of Cowen-Douglas Operators

  Published:2004-08-01
 Printed: Aug 2004
  • Chunlan Jiang
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Abstract

Let $\cal H$ be a complex separable Hilbert space and ${\cal L}({\cal H})$ denote the collection of bounded linear operators on ${\cal H}$. An operator $A$ in ${\cal L}({\cal H})$ is said to be strongly irreducible, if ${\cal A}^{\prime}(T)$, the commutant of $A$, has no non-trivial idempotent. An operator $A$ in ${\cal L}({\cal H})$ is said to a Cowen-Douglas operator, if there exists $\Omega$, a connected open subset of $C$, and $n$, a positive integer, such that (a) ${\Omega}{\subset}{\sigma}(A)=\{z{\in}C; A-z {\text {not invertible}}\};$ (b) $\ran(A-z)={\cal H}$, for $z$ in $\Omega$; (c) $\bigvee_{z{\in}{\Omega}}$\ker$(A-z)={\cal H}$ and (d) $\dim \ker(A-z)=n$ for $z$ in $\Omega$. In the paper, we give a similarity classification of strongly irreducible Cowen-Douglas operators by using the $K_0$-group of the commutant algebra as an invariant.
MSC Classifications: 47A15, 47C15, 13E05, 13F05 show english descriptions Invariant subspaces [See also 47A46]
Operators in $C^*$- or von Neumann algebras
Noetherian rings and modules
Dedekind, Prufer, Krull and Mori rings and their generalizations
47A15 - Invariant subspaces [See also 47A46]
47C15 - Operators in $C^*$- or von Neumann algebras
13E05 - Noetherian rings and modules
13F05 - Dedekind, Prufer, Krull and Mori rings and their generalizations
 

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