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.