Approximation and Similarity Classification of Stably Finitely Strongly Irreducible Decomposable Operators

Let 𝓗 be a complex separable Hilbert space and ℒ(𝓗) denote the collection of bounded linear operators on 𝓗. In this paper, we show that for any operator A ∈ ℒ(𝓗), there exists a stably finitely (SI) decomposable operator A∈, such that ‖A−A∈‖ 𝓗 ∈ andA′(A∈)/ rad A′(A∈) is commutative, where rad A′(A∈) is the Jacobson radical of A′(A∈). 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.