Backward Shift and Nearly Invariant Subspaces of Fock-type Spaces

We study the structure of the backward shift invariant and nearly invariant subspaces in weighted Fock-type spaces F-W(p), whose weight is not necessarily radial. We show that in the spaces 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 P-n, that is, 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 that show that the result fails for general Fock-type spaces of larger growth.

Automated summary

The study examines the structure of backward shift invariant and nearly invariant subspaces in weighted Fock-type spaces. It is shown that in spaces containing polynomials, nontrivial backward shift invariant subspaces have a specific form, whereas nearly invariant subspaces have a more complex structure. An analogue of de Branges' ordering theorem is established for spaces of slow growth, but the result does not hold for general Fock-type spaces of larger growth.