On the similarity of restriction of the operator to an invariant subspace

Let $M_z$ be the multiplication operator on the Bergman space and $M_I$ denote the restriction of $M_z$ to an invariant subspace $I$. A question raised by K. Zhu is that when are two restriction operators $M_I$ and $M_J$ are similar? In this note, we give some sufficient conditions of this problem in a general case.