Bergman inner functions and $m$-hypercontractions

Abstract

Let $H_m(\mathbb B,\mathcal D)$ be the $\mathcal D$-valued functional Hilbert space with reproducing kernel $K_m(z,w) = (1-\langle z,w\rangle)^{-m}1_{\mathcal D}$. A $K_m$-inner function is by definition an operator-valued analytic function $W: \mathbb B \rightarrow L(\mathcal E, \mathcal D)$ such that $\|Wx\|_{H_m(\mathbb B,\mathcal D)} = \|x\|$ for all $x \in \mathcal E$ and $(W\mathcal E) \perp M_z^{\alpha}(W\mathcal E)$ for all $\alpha \in \mathbb N^n \setminus \{0\}$. We show that the $K_m$-inner functions are precisely the functions of the form $W(z) = D + C \sum^m_{k=1}(1 - ZT^*)^{-k}ZB$, where $T \in L(H)^n$ is a pure $m$-hypercontraction and the operators $T^*, B, C,D$ form a $2 \times 2$-operator matrix satisfying suitable conditions. Thus we extend results proved by Olofsson on the unit disc to the case of the unit ball $\mathbb B \subset \mathbb C^n$.