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Similarity of restriction of the operator to an invariant subspace

The Bergman space \$L^2_a(\mathbb{D})\$ is the closed subspace of \$L^2(\mathbb{D},dA)\$ consisting of analytic functions, where \$dA\$ is the normalized area measure on \$\mathbb{D}\$. For an invariant subspace \$I\$ denote the restriction of \$M_z\$to \$I\$. An open problem asked by K. Zhu is that when are two restriction operators \$M_I\$and \$M_J\$ are similar? In this talk, we give some sufficient conditions in a general case.