Multivariable generalizations of the Schur

class: positive kernel characterization and transfer function realization Joseph A. Ball, Animikh Biswas, Quanlei Fang and Sanne ter Horst Abstract. The operator-valued Schur-class is defined to be the set of holo-morphic functions S mapping the unit disk into the space of contraction op-erators between two Hilbert spaces. There are a number of alternate charac-terizations: the operator of multiplication by S defines a contraction operator between two Hardy Hilbert spaces, S satisfies a von Neumann inequality, a certain operator-valued kernel associated with S is positive-definite, and S can be realized as the transfer function of a dissipative (or even conservative) discrete-time linear input/state/output linear system. Various multivariable generalizations of this class have appeared recently, one of the most encom-passing being that of Muhly and Solel where the unit disk is replaced by the strict unit ball of the elements of a dual correspondence Eσ associated with a W ∗-correspondence E over a W ∗-algebra A together with a ∗-representation σ of A. The main new point which we add here is the introduction of the notion of reproducing kernel Hilbert correspondence and identification of the Muhly-Solel Hardy spaces as reproducing kernel Hilbert correspondences as-sociated with a completely positive analogue of the classical Szego ̈ kernel. In this way we are able to make the analogy between the Muhly-Solel Schur class and the classical Schur class more complete. We also illustrate the theory by specializing it to some well-studied special cases; in some instances there result new kinds of realization theorems