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Approximation and Similarity Classification of Stably Finitely Strongly Irreducible Decomposable Operators

Published online by Cambridge University Press:  20 November 2018

He Hua ,
Dong Yunbai  and
Guo Xianzhou
He Hua
Affiliation:
Department of Mathematics, Hebei University of Technology, Tianjin 300130, P.R. China, e-mail: [email protected], [email protected], [email protected]
Dong Yunbai
Affiliation:
Department of Mathematics, Hebei University of Technology, Tianjin 300130, P.R. China, e-mail: [email protected], [email protected], [email protected]
Guo Xianzhou
Affiliation:
Department of Mathematics, Hebei University of Technology, Tianjin 300130, P.R. China, e-mail: [email protected], [email protected], [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 $\mathcal{H}$. In this paper, we show that for any operator $A\,\in \,\mathcal{L}\left( \mathcal{H} \right)$, there exists a stably finitely $\left( \text{SI} \right)$ decomposable operator ${{A}_{\epsilon }}$, such that $\left\| A-{{A}_{\epsilon }} \right\|\,<\,\epsilon$ and ${{\mathcal{A}}^{\prime }}\left( {{A}_{\epsilon }} \right)/\text{rad}\,{{\mathcal{A}}^{\prime }}\left( {{A}_{\epsilon }} \right)$ is commutative, where rad ${{\mathcal{A}}^{\prime }}\left( {{A}_{\epsilon }} \right)$ is the Jacobson radical of ${{\mathcal{A}}^{\prime }}\left( {{A}_{\epsilon }} \right)$. Moreover, we give a similarity classification of the stably finitely decomposable operators that generalizes the result on similarity classification of Cowen–Douglas operators given by C. L. Jiang.

Keywords

47A0547A5546H20K0-groupstrongly irreducible decompositionCowen–Douglas operatorscommutant algebrasimilarity classification
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 2 , 01 April 2010 , pp. 305 - 319
Copyright
Copyright © Canadian Mathematical Society 2010

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