Multivariable Bergman shifts and Wold decompositions

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.