Treffer: Nearly Invariant Subspaces Related to Multiplication Operators in Hilbert Spaces of Analytic Functions: Nearly invariant subspaces related to multiplication operators in Hilbert spaces of analytic functions

Let \({\mathcal H}\) be a Hilbert space of analytic functions defined on an open subset \({\mathcal W}\) of \({\mathbb C}^d\), stable under the multiplication operator \(M_u\) induced by some function \(u.\) Given a subspace \({\mathcal M}\) of \({\mathcal H}\) which is ``nearly invariant under division by \(u\)'', this paper provides a factorization linking each element of \({\mathcal M}\) to elements of \({\mathcal M}\ominus(M\cap M_u{\mathcal H})\) on the inverse image under \(u\) of a certain complex disc, for which a relatively simple formula is given. As its application, some interesting results involving an \(H^2\) control are also obtained, in particular, a factorization for the kernel of Toeplitz operators on Dirichlet spaces and a localization for the problem of extraneous zeros.