Unitary Equivalence, Similarity and Calculation of K ₀–Group

Abstract

In this paper, we are concerned with the classi.cation of operators on complex separable Hilbert spaces, in the unitary equivalence sense and the similarity sense, respectively. We show that two strongly irreducible operators A and B are unitary equivalent if and only if W*(A⊕B)'≈ M ₂(C), and two operators A and B in ℬ₁(Ω) are similar if and only if ${{\fancyscript A}}'(A \oplus B)/J \approx M_{2} (C).$Moreover, we obtain V (H ^∞(Ω, μ)) ≈ N and K ₀(H ^∞(Ω, μ))≈ Z by the technique of complex geometry, where Ω is a bounded connected open set in C, and μ is a completely non–reducing measure on $\bar{\Omega }$.