On the structure of invariant subspaces for the shift operator on Bergman spaces

TL;DR

This paper investigates the structure of invariant subspaces for the shift operator on Bergman spaces, revealing that each nonzero invariant subspace contains elements outside the Hardy space, thus providing insight into their complex structure.

Contribution

It demonstrates that every nonzero invariant subspace of the shift operator on Bergman spaces contains elements outside the Hardy space, offering a new structural perspective.

Findings

Every nonzero invariant subspace contains elements outside the Hardy space

Provides a partial description of the structure of invariant subspaces

Highlights the complexity of invariant subspaces in Bergman spaces

Abstract

It is well known that the structure of nontrival invariant subspaces for the shift operator on the Bergman space is extremely complicated and very little is known about their specific structures, and that a complete description of the structure seems unlikely. In this paper, we find that any invariant subspace $M(\neq \{0\})$ for the shift operator $M_z$ on the Bergman space $A^{^p}_\alpha(\D)~(1\leq p< \infty,~~-1<\alpha<\infty)$ contains a nonempty subset that lies in $A^{^p}_\alpha(\D)\backslash H^{^p}(\D)$. To a certain extent, this result describes the specific structure of every nonzero invariant subspace for the shift operator $M_z$ on $A^{^p}_\alpha(\D)$.