Backward shift and nearly invariant subspaces of Fock-type spaces

TL;DR

This paper investigates the structure of backward shift and nearly invariant subspaces in weighted Fock-type spaces, revealing that in certain cases these subspaces are finite-dimensional polynomial spaces, while in others the structure is more complex.

Contribution

It characterizes backward shift invariant subspaces in weighted Fock spaces and extends de Branges' Ordering Theorem to spaces of slow growth, highlighting differences in larger growth spaces.

Findings

Backward shift invariant subspaces are finite-dimensional polynomial spaces in certain Fock spaces.

An analogue of de Branges' Ordering Theorem is established for spaces of slow growth.

Counterexamples show the structure differs in larger growth Fock spaces.

Abstract

We study the structure of the backward shift invariant and nearly invariant subspaces in weighted Fock-type spaces $\mathcal{F}_W^p$, whose weight $W$ is not necessarily radial. We show that in the spaces $\mathcal{F}_W^p$ which contain the polynomials as a dense subspace (in particular, in the radial case) all nontrivial backward shift invariant subspaces are of the form $\mathcal{P}_n$, i.e., finite dimensional subspaces consisting of polynomials of degree at most $n$. In general, the structure of the nearly invariant subspaces is more complicated. In the case of spaces of slow growth (up to zero exponential type) we establish an analogue of de Branges' Ordering Theorem. We then construct examples which show that the result fails for general Fock-type spaces of larger growth.