Some sufficient conditions for the division property of invariant subspaces in weighted Bergman spaces (Q676207)

The authors continue the study of the invariant subspaces of the multiplication operator \(f\mapsto zf\) on weighted Bergman spaces \(L^p(\Omega,\mu)\cap H(\Omega)\). In the paper under review they are interested in invariant subspaces \({\mathcal M}\) having the division property, a property being, in most cases, equivalent to \(\dim{\mathcal M}/(z-\lambda){\mathcal M}=1\). The authors look for sufficient conditions in terms of the local boundary behaviour of the functions in \({\mathcal M}\) which ensure that \({\mathcal M}\) has the division property. In the more concrete setting of the weighted Bergman spaces \(A_\alpha^p(\mathbb{D})\), \(1\leq p<\infty\), \(\alpha>-1\), of all analytic functions on the unit disk \(\mathbb{D}\) for which \[ |f|_p^p= \int_{\mathbb{D}}|f|^p(1-|z|)^\alpha dA<\infty, \] they obtain that if \({\mathcal M}\) contains a function that is locally Nevanlinna near a boundary point of \(\mathbb{D}\), then \({\mathcal M}\) has the division property. In the case of the unweighted Bergman space \(L_a^p(\mathbb{D}):= A_0^p\) the authors show the following result: Let \(f,g\in L_a^p(\mathbb{D})\) and let \([f]\) denote the smallest invariant subspace of \(L_a^p(\mathbb{D})\) that contains \(f\). Then the closed linear span of \([f]\) and \([g]\) in \(L_a^p(\mathbb{D})\) has the division property if there exist a boundary point \(\alpha\) of \(\mathbb{D}\), a neighborhood \(V\) of \(\alpha\) and positive reals \(s\) and \(r\) with \(1/r+ 1/s= 1/p\) for which \(f\in L^r(V\cap\mathbb{D})\) and \(g\in L^s(V\cap\mathbb{D})\).