On the similarity of operators restricted 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 in [44] is when two restriction operators $M_I$ and $M_J$ are similar. Influenced by this open question, this note considers a more general case. Specifically, the multiplication operator on the Bergman space is replaced by the adjoint of Cowen-Douglas operators, and a direct sum of the multiplication operator acting on certain analytic functional Hilbert spaces. Furthermore, we give some sufficient conditions for these questions involving complex geometric objects of Hermitian holomorphic vector bundles.