Strongly irreducible decomposition and similarity classification of operators

Abstract

Let H be a complex separable Hilbert space and let L(H) denote the collection of bounded linear operators on H. In this paper, we show that if T = A (n1) 1 ⊕ A (n2) 2 ⊕ · · · ⊕ A (nk) k , where Ai 6∼ Aj for 1 ≤ i 6= j ≤ k, andA′(Ai)/ radA′(Ai) is commutative, K0(A′(Ai)) ∼= Z for i = 1, 2, . . . , k, and for any positive integer n and minimal idempotent P ∈ A′(T (n)), A′(T |PH(n) )/ radA ′(T |PH(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 K0-group of the commutant algebra as an invariant.