  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.
