ALGEBRAIC EQUIVALENCE OF QUASINORMAL OPERATORS

Let $T_j =N_j\oplus( S\otimes A_j)$ be quasinormal, where $N_j$ is normal and $A_j$ is a positive definite operator, $j = 1, 2$. We show that $T_1$ is algebraically equivalent to $T_2$ if and only if $\sigma(A_1) =\sigma(A_2)$ and $\sigma(N_1)\backslash\sigma_{ap}(S\otimes A_1) =\sigma(N_2)\backslash\sigma_{ap}(S\otimes A_2)$. This generalizes the corresponding result for normal and isometric operators.