Hypercontractions and factorizations of multipliers in one and several variables

Abstract: We introduce the notion of characteristic functions for commuting tuples of hypercontractions on Hilbert spaces, as a generalization of the notion of Sz.-Nagy and Foias characteristic functions of contractions. We present an explicit method to compute characteristic functions of hypercontractions and relate characteristic functions by means of the factors of Schur-Agler class of functions and universal multipliers on the unit ball in $\mathbb{C}^n$. We also offer some factorization properties of multipliers. Characteristic functions of hypercontrctions are complete unitary invariant. The Drury-Arveson space and the weighted Bergman spaces on the unit ball continues to play a significant role in our consideration. Our results are new even in the special case of single hypercontractions.