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 $K_m(z,w)=(1-⟨z,w⟩{)}^{-m}$. 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.