Let H be a complex separable Hilbert space and L(H) denote the collection of bounded linear operators onH. An operator A in L(H) is said to be strongly irreducible, if A' (T), the commutant of A, has no non-trivial idempotent. An operator A in L(H) is said to be a Cowen-Douglas operator, if there exists Ω, a connected open subset of C, and n, a positive integer, such that

We prove that if T is a biquasitriangular operator on a Hilbert space H with connected spectrum then T may be approximated by a strongly irreducible operator S with S−T compact and small.

This paper shows that every operator which is quasisimilar to strongly irreducible Cowen-Douglas operators is still strongly irreducible. This result answers a question posted by Davidson and Herrero (ref. [1]).

A bounded linear operator T on the Hilbert space H is called strongly irreducible if T does not commute with any nontrivial idempotentoperator. One says that T has a finite (SI) decomposition if T can be writtenas the direct sum of finitely many strongly irreducible operators. In this paper, we use the Ko-group of the commutant of operators to characterize operators with unique finite (SI) decomposition up to similarity. Also weshow that the Ko-group of H∞(Ω) is isomorphic to the integers, where is simply connected.