Nearly invariant subspaces in spaces of entire functions

A subspace $H_0$ of a Hilbert space $H$ consisting of functions analytic in some domain is said to be nearly invariant if the function $f(z)/(z-w)$ belongs to $H_0$ whenever $f \in H_0$ and $f(w)=0$. In the talk we discuss the structure problem for nearly invariant subspaces of the Fock-type spaces of entire functions. The talk is based on a joint work with Alexandru Aleman (Lund University) and Yurii Belov (St. Petersburg State University).