Multivariable Bergman shifts and Wold decompositions

TL;DR

This paper characterizes certain multivariable operators on Hilbert spaces using algebraic identities, extending classical Wold decompositions to the unit ball setting for multivariable Bergman shifts.

Contribution

It provides a new algebraic characterization of multivariable Bergman shifts and extends the classical Wold decomposition to the unit ball in several complex variables.

Findings

Characterization of commuting row contractions decomposing into spherical coisometries and multiplication tuples.

Extension of Wold decomposition to multivariable settings on the unit ball.

Reduction to classical Wold decomposition in special cases.

Abstract

Let $H_m(\mathbb B)$ be the analytic functional Hilbert space on the unit ball $\mathbb B \subset \mathbb C^n$ with reproducing kernel $K_m(z,w) = (1 - \langle z,w \rangle)^{-m}$. Using algebraic operator identities we characterize those commuting row contractions $T \in L(H)^n$ on a Hilbert space $H$ that decompose into the direct sum of a spherical coisometry and copies of the multiplication tuple $M_z \in L(H_m(\mathbb B))^n$. For $m=1$, this leads to a Wold decomposition for partially isometric commuting row contractions that are regular at $z = 0$. For $m = 1 = n$, the results reduce to the classical Wold decomposition of isometries. We thus extend corresponding one-variable results of Giselsson and Olofsson to the case of the unit ball.