Characterizations of functions in wandering subspaces of the Bergman Shift via the Hardy space of the Bidisc

TL;DR

This paper characterizes functions in wandering subspaces of the Bergman shift using the Hardy space of the bidisc, providing necessary and sufficient conditions, and explores related invariant subspace properties.

Contribution

It offers a new functional and coefficient characterization of wandering subspace functions and establishes a decomposition theorem related to the Bergman shift's universal property.

Findings

Characterization of wandering subspace functions in the Bergman space.

Conditions for subspaces to be wandering subspaces.

Decomposition theorem for operators between wandering subspaces.

Abstract

Let $\mathcal{W}$ be the corresponding wandering subspace of an invariant subspace of the Bergman shift. By identifying the Bergman space with $H^2(\mathbb{D}^2)\ominus[z-w]$, a sufficient and necessary conditions of a closed subspace of $H^2(\mathbb{D}^2)\ominus[z-w]$ to be a wandering subspace of an invariant subspace is given also, and a functional charaterization and a coefficient characterization for a function in a wandering subspace are given. As a byproduct, we proved that for two invariant subspaces $\mathcal{M}$, $\mathcal{N}$ with $\mathcal{M}\supsetneq\mathcal{N}$ and $dim(\mathcal{N}\ominus B\mathcal{N})<\infty$ $dim(\mathcal{M}\ominus B\mathcal{M})=\infty$, then there is an invariant subspace $\mathcal{L}$ such that $\mathcal{M}\supsetneq\mathcal{L}\supsetneq\mathcal{N}$. Finally, we define an operator from one wandering subspace to another, and get a decomposition theorem for such an operator which is related to the universal property of the Bergman shift.