The eigenvalues, numerical ranges, and invariant subspaces of the Bergman Toeplitz operators over the bidisk (Q6549804)

Let \(L^\infty(\mathbb D^2)\) be the set of bounded functions on the bidisk \(\mathbb D^2\), \(H^\infty(\mathbb D^2)\) be the set of bounded holomorphic functions on \(\mathbb D^2\). Let \(z\in \mathbb D^2\), we denote by \(K_z\) the reproducing kernel of the Hilbert space \(L_a^2(\mathbb D^2)\), the space of square-integrable analytic functions on \(\mathbb D^2\), and \(T_\varphi\) be the Toeplitz operator defined on \(L_a(\mathbb D^2)\) with symbol \(\varphi\in L^\infty(\mathbb D^2)\). Then, the authors provide a partial answer to the following question: For \(\varphi\in L^\infty(\mathbb D^2)\), if there exists some \(z\in\mathbb D^2\) such that \(K_z\) is an eigenvector of \(T_\varphi\) on \(L_a(\mathbb D^2)\), is it necessary that \(\varphi\in\overline{H^\infty(\mathbb D^2)}\)?\N\NFurthermore, they provide an affirmative answer of the following two questions:\N\N\(\bullet\) Let \(A\) be a bounded linear operator on \(L^2_a(\mathbb D^2)\). If for any \(z\in\mathbb D^2\), \(K_z\) is always an eigenvector of \(A\), then does there exist any \(\varphi\in H^\infty(\mathbb D^2)\) such that \(A = T_\varphi\)?\N\N\(\bullet\) For \(\varphi\in L^\infty(\mathbb D^2)\) being a 2-harmonic function, could we give a description for the numerical range of the Bergman Toeplitz operator \(T_\varphi\) over the bidisk by the range of its symbol?