Pure contractive multipliers of some reproducing kernel Hilbert spaces and applications

Abstract

A contraction T on a Hilbert space H is said to be pure if the sequence {T * n } n converges to 0 in the strong operator topology. In this article, we prove that for contractions T , which commute with certain tractable tuples of commuting operators X = (X 1 , . . . , X n ) on H, the following statements are equivalent:(i) T is a pure contraction on H, (ii) the compression P W(X) T | W(X) is a pure contraction, where W(X) is the wandering subspace corresponding to the tuple X.An operator-valued multiplier Φ of a vector-valued reproducing kernel Hilbert space (rkHs) is said to be pure contractive if the associated multiplication operator M Φ is a pure contraction. Using the above result, we find that operator-valued mulitpliers Φ(z) of several vector-valued rkHs's on the polydisc D n as well as the unit ball B n in C n are pure contractive if and only if Φ(0) is a pure contraction on the underlying Hilbert space. The list includes Hardy, Bergman and Drury-Arveson spaces. Finally, we present some applications of our characterization of pure contractive multipliers associated with the polydisc.