K-inner functions and K-contractions

TL;DR

This paper characterizes $K$-inner functions in the context of unitarily invariant kernels on the unit ball, introduces a transfer function realization, and extends classical dilation theorems to a broader setting.

Contribution

It provides a new characterization of $K$-inner functions via transfer functions and generalizes Arveson's uniqueness theorem for $K$-dilations.

Findings

Characterization of $K$-inner functions as transfer functions.

Construction of a canonical $K$-inner function for $K$-contractions.

Extension of Arveson's uniqueness theorem to the $K$-contraction setting.

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.