Nearly invariant subspaces related to multiplication operators in Hilbert spaces of analytic functions

摘要

We consider Hilbert spaces H of analytic functions defined on an open subset W of C-d, stable under the operator M,, of multiplication by some function u. Given a subspace M of R which is "nearly invariant under division by u", we provide a factorization linking each element of M to elements of M circle minus (M boolean AND M-u H) on the inverse image under u of a certain complex disc, for which we give a relatively simple formula. By applying these results to W = D and u(z) = z, we obtain interesting results involving a H-2 -norm control. In particular, we deduce a factorization for the kernel of Toeplitz operators on Dirichlet spaces. Finally, we give a localization for the problem of extraneous zeros.