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Similarity Classification of Cowen-Douglas Operators

Published online by Cambridge University Press:  20 November 2018

Chunlan Jiang
Chunlan Jiang*
Affiliation:
Department of Mathematics, Hebei Normal University, Shijiazhuang 050016, P.R. China e-mail: [email protected]
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Abstract

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Let $\mathcal{H}$ be a complex separable Hilbert space and $\mathcal{L}\left( \mathcal{H} \right)$ denote the collection of bounded linear operators on $\mathcal{H}$. An operator $A$ in $\mathcal{L}\left( \mathcal{H} \right)$ is said to be strongly irreducible, if ${{\mathcal{A}}^{\prime }}(T)$, the commutant of $A$, has no non-trivial idempotent. An operator $A$ in $\mathcal{L}\left( \mathcal{H} \right)$ is said to be a Cowen-Douglas operator, if there exists $\Omega $, a connected open subset of $C$, and $n$, a positive integer, such that

  1. (a)

    $$\Omega \,\subset \,\sigma (A)\,=\,\left\{ z\,\in \,C|\,A-z\,\text{not}\,\text{invertible} \right\};$$

  2. (b)

    $$\text{ran(A}-z\text{)}\,\text{=}\,\mathcal{H}\text{,}\,\text{for}\,z\,\text{in}\,\Omega \text{;}$$

  3. (c)

    $${{\vee }_{z\in \Omega }}\,\ker (A-\,z)\,=\,\mathcal{H}\,\text{and}$$

  4. (d)

    $$\dim\,\ker (A-z)\,=\,n\,\text{for}\,z\,\text{in}\,\Omega $$

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.

Keywords

47A1547C1513E0513F05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 4 , 01 August 2004 , pp. 742 - 775
Copyright
Copyright © Canadian Mathematical Society 2004

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