Multivariable Bergman shifts and Wold decompositions

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Let Hₘ(B) be the analytic functional Hilbert space on the unit ball B ⊂ Cⁿ with reproducing kernel Kₘ(z,w) = (1 - z,w)⁻ᵐ. Using algebraic operator identities we characterize those commuting row contractions T ∈ L(H)ⁿ on a Hilbert space H that decompose into the direct sum of a spherical coisometry and copies of the multiplication tuple Mz ∈ L(Hₘ(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.