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# Approximation and Similarity Classification of Stably Finitely Strongly Irreducible Decomposable Operators

Published:2009-12-04
Printed: Apr 2010
• He Hua
• Dong Yunbai
• Guo Xianzhou
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## Abstract

Let $\mathcal H$ be a complex separable Hilbert space and ${\mathcal L}({\mathcal H})$ denote the collection of bounded linear operators on ${\mathcal H}$. In this paper, we show that for any operator $A\in{\mathcal L}({\mathcal H})$, there exists a stably finitely (SI) decomposable operator $A_\epsilon$, such that $\|A-A_{\epsilon}\|<\epsilon$ and ${\mathcal{\mathcal A}'(A_{\epsilon})}/\operatorname{rad} {{\mathcal A}'(A_{\epsilon})}$ is commutative, where $\operatorname{rad}{{\mathcal A}'(A_{\epsilon})}$ is the Jacobson radical of ${{\mathcal A}'(A_{\epsilon})}$. 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: $K_{0}$-group, strongly irreducible decomposition, Cowen—Douglas operators, commutant algebra, similarity classification
 MSC Classifications: 47A05 - General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47A55 - Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15] 46H20 - Structure, classification of topological algebras