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 $m = 1 = n$, one obtains the classical Wold decomposition of isometries. To prove the above results we extend a corresponding one-variable Wold-type decomposition theorem of Giselsson and Olofsson (Complex Anal Oper Theory 6:829–842, 2012) to the case of the unit ball.