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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 $K_{0}$-group, strongly irreducible decomposition, Cowen—Douglas operators, commutant algebra, similarity classification
MSC Classifications: 47A05, 47A55, 46H20 show english descriptions General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15]
Structure, classification of topological algebras
47A05 - General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A55 - Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15]
46H20 - Structure, classification of topological algebras
 

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