Realizations via Preorderings with Application to the Schur Class

Michael A. Dritschel

Published 2015-03-04Version 1

We extend Agler's notion of a function space defined in terms of test functions to include products, in analogy with practices in real algebraic geometry. This is done over abstract sets and no additional property, such as analyticity, is assumed. We prove a realization theorem for functions in the unit ball of such an algebra, otherwise known as the Schur-Agler class. Restricting to the context of so-called ample and nearly ample preorderings, the realization theorem can be further strengthened, enough so as to allow applications to, among other things, Pick type interpolation problems. This is achieved through the construction of matrix valued auxiliary test functions. When the domain is the polydisk $D^d$, the algebras of functions obtained include $H^\infty(\mathbb D^d,\mathcal{L(H)})$ and $A(\mathbb D^d,\mathcal{L(H)})$, the multivariable analogue of the disk algebra. We show that a restricted class of representations called Brehmer representations are completely contractive (for representations of $H^\infty(\mathbb D^d,\mathcal{L(H)})$ we must also assume weak continuity). These include as a subclass those (weakly continuous) representations which are contractive on the auxiliary test functions. As a consequence it is proved that over the polydisk $D^d$, (weakly continuous) representations which are $2^{d-2}$ contractive are completely contractive. In particular, the generators of such a representation, which are commuting contractions, have a commuting unitary dilation.