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 $K_m(z,w)=(1-⟨z,w⟩{)}^{-m}{1}_{\mathcal{D}}$. A $K_m$-inner function is by definition an operator-valued analytic function $W:\mathbb{B}\to L(\mathcal{E},\mathcal{D})$ such that $\Vert Wx{\Vert }_{H_m(\mathbb{B},\mathcal{D})}=\Vert x\Vert$ 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 $W(z)=D+C{\sum }_{k=1}^m(1-Z{T}^{\ast }{)}^{-k}ZB$, where $T\in L(H{)}^n$ is a pure $m$-hypercontraction and the operators $T^{\ast },B,C,D$ 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 $\mathbb{B}\subset {\mathbb{C}}^n$.