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Approximation and Similarity Classification of Stably Finitely Strongly Irreducible Decomposable Operators
Published online by Cambridge University Press: 20 November 2018
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.
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