Reducing subspaces for Toeplitz operators on the polydisk (Q2857011)

A closed subspace \({\mathcal M}\) of a Hilbert space is called a reducing subspace of an operator \(T\) if \(T({\mathcal M})\subset {\mathcal M}\) and \(T^*({\mathcal M})\subset{\mathcal M}\). A nontrivial reducing subspace \({\mathcal M}\) of \(T\) is said to be minimal if the only reducing subspaces of \(T\) containing \({\mathcal M}\) are \({\mathcal M}\) and \(\{0\}\). In this note, the authors completely characterize the reducing subspaces of the Toeplitz operator \(T_\varphi\) with symbol \(\varphi(z)=z_1^Nz_2^M\) on the weighted Bergman spaces \(A_\alpha^2({\mathbb D}^2)\) on the polydisk \({\mathbb D}^2\), where \(\alpha>-1\) and \(M,N\) are positive integers with \(N\neq M\). In particular, it is shown that the minimal reducing subspaces of \(T_\varphi\) on the weighted Bergman spaces \(A_\alpha^2({\mathbb D}^2)\) with \(\alpha\neq 0\) and on the unweighted Bergman space \(A^2({\mathbb D}^2)\) are different.