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 ℬ_{1}(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
$$
\bar{\Omega }
$$
.