Quasi-wandering subspaces in the Bergman space
Let \(T\) be a bounded linear operator on a complex separable Hilbert space \(H\) and let \(M\subset H\) be a subspace invariant under \(T\). In the paper under review, \(P_MT(H\ominus M)\) is called the quasi-wandering subspace for \(M\). Here, \(P_M\) means the orthogonal projection onto \(M\). For the unilateral shift \(S\) on the Hardy space on the open unit disk \(\mathbb D\), a quasi-wandering subspace coincides with a wandering subspace \(M\ominus SM\). The authors investigate quasi-wandering subspaces of the shift \(\mathcal Bf=zf\) on the Bergman space \(L_a^2(\mathbb D)\). They prove that, if \(I\) is a nontrivial invariant subspace for \(\mathcal B\), then \(P_I\mathcal B(L_a^2\ominus I)\) is not dense in \(I\) if and only if \(I\cap\mathcal D\neq\{0\}\), where \(\mathcal D=\{f: f\) is analytic in \(\mathcal D\) with \(f\in L_a^2(\mathbb D)\}\) is the Dirichlet space.
scientific article; zbMATH DE number 5774007
scientific article