  An interesting and recently much studied generalization of the classical
Schur class is the class of contractive operator-valued multipliers for the
reproducing kernel Hilbert space ${\mathcal H}(k_{d})$ on the unit ball
${\mathbb B}^{d} \subset {\mathbb C}^{d}$, where $k_{d}$ is the positive kernel
$k_{d}(\lambda, \zeta) = 1/(1 - < \lambda, \zeta >)$ on ${\mathbb B}^{d}$. We
study this space from the point of view of realization theory and functional
models of de Branges-Rovnyak type. We highlight features which depart from the
classical univariate case: coisometric realizations have only partial
uniqueness properties, the nonuniqueness can be described explicitly, and this
description assumes a particularly concrete form in the functional-model
context.
