K-inner functions and K-contractions

Abstract

For a large class of unitarily invariant reproducing kernel functions $K$ on the unit ball ${\mathbb{B}}_d$ in ${\mathbb{C}}^d$, we characterize the $K$-inner functions on ${\mathbb{B}}_d$ as functions admitting a suitable transfer function realization. We associate with each $K$-contraction $T\in L(H{)}^d$ a canonical operator-valued $K$-inner function and extend a uniqueness theorem of Arveson for minimal $K$-dilations to our setting. We thus generalize results of Olofsson for $m$-hypercontractions on the unit disc and of the first named author for $m$-hypercontractions on the unit ball.