Strongly irreducible decomposition and similarity classification of operators

Abstract

Let $\mathcal H$ be a complex separable Hilbert space and let ${\mathcal L}({\mathcal H})$ denote the collection of bounded linear operators on ${\mathcal H}$. In this paper, we show that if $T=A^{(n_1)}_{1}\oplus A^{(n_2)}_{2}\oplus\dots\oplus A^{(n_k)}_{k}$, where $A_{i}\not\sim A_{j}$ for $1\leq i\neq j\leq k$, and ${\mathcal A}'(A_{i})/\rad{\mathcal A}'(A_{i})$ is commutative, $K_{0}({\mathcal A}'(A_{i}))\cong Z$ for $i=1,2,\dots, k$, and for any positive integer $n$ and minimal idempotent $P\in {\mathcal A}'(T^{(n)})$, ${\mathcal A}'(T^{(n)}|_{P{\mathcal H}^{(n)}})/\rad{\mathcal A}'(T^{(n)}|_{P{\mathcal H}^{(n)}})$ is commutative, then $T$ is a stably finitely decomposable operator and has a stably unique (SI) decomposition up to similarity. Moreover, we give a similarity classification of the operators which satisfy the above conditions by using the $K_{0}$-group of the commutant algebra as an invariant.