Bergman inner functions and m-hypercontractions

Original · EN

Let Hₘ(B,D) be the D-valued functional Hilbert space with reproducing kernel Kₘ(z,w) = (1- z,w)⁻ᵐ1D. A Kₘ-inner function is by definition an operator-valued analytic function W: B → L(E, D) such that Wxₕₘ₍B,D₎ = x for all x ∈ E and (WE) ⊥ Mzα(WE) for all α∈ Nⁿ {0}. We show that the Kₘ-inner functions are precisely the functions of the form W(z) = D + C ∑ᵐₖ₌₁(1 - ZT*)⁻ᵏZB, where T ∈ L(H)ⁿ is a pure m-hypercontraction and the operators T*, B, C,D form a 2 × 2-operator matrix satisfying suitable conditions. Thus we extend results proved by Olofsson on the unit disc to the case of the unit ball B ⊂ Cⁿ.