Transfer-function realization for multipliers of the Arveson space

Abstract

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 H ( k d ) on the unit ball B d ⊂ C d , where k d is the positive kernel k d ( λ , ζ ) = 1 / ( 1 − 〈 λ , ζ 〉 ) on 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.