Nearly Invariant Subspaces Related to Multiplication Operators in Hilbert Spaces of Analytic Functions

Abstract.

We consider Hilbert spaces $\mathcal{H}$ of analytic functions defined on an open subset $\mathcal{W}$ of $\mathbb{C}^d$ , stable under the operator M _u of multiplication by some function u. Given a subspace $\mathcal{M}$ of $\mathcal{H}$ which is ”nearly invariant under division by u”, we provide a factorization linking each element of $\mathcal{M}$ to elements of $\mathcal{M}\ominus (\mathcal{M} \cap M_u \mathcal{H})$ on the inverse image under u of a certain complex disc, for which we give a relatively simple formula. By applying these results to $\mathcal{W} = \mathbb{D}$ and u(z)  =  z, we obtain interesting results involving a H²-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.