Finite-rank intermediate Hankel operators on the Bergman space (Q5932800)

Let \(L^2_a=L^2(D, d\mu)\cap Hol (D)\) be the weighted (closed) Bergman space on the unit disk and let \(P^M\) be the orthogonal projection of \(L^2\) onto closed subspace \(M\), such that \(zM\subseteq M\) and \(M \supseteq zL^2_a\). The authors consider so-called intermediate Hankel operators \(H_\phi^M f=(I-P^ M)(\phi f), \phi \in L^\infty, f \in L^2_a.\) Such operators are intermediate between big \(H_\phi^{\text{big}}\) and small \(H_\phi^{\text{small}}\) Hankel operators, where \(H_\phi^{\text{big}} f=(I-P)(\phi f), H_\phi^{\text{small}} f=(I-Q)(\phi f), \phi \in L^\infty, f \in L^2_a\) and \(P, Q\) are orthogonal projections of \(L^2\) onto \(L^2_a\) and \({\widetilde L}^2_{a,0}=\{g\in L^2, \widetilde{g}\in L^2_a, g(0)=0\}\) respectively. The first projection of them corresponds to the case \(M=L^2_a\) and \(H_\phi^{\text{big}}=0\) if it is a finite-rank operator. It is studied when the intermediate Hankel operators (in particular, and \(H_\phi^{\text{small}}\)) are finite-rank operators in the weighted Bergman in depend on \(M\).