Pure contractive multipliers of some reproducing kernel Hilbert spaces and applications

TL;DR

This paper characterizes pure contractive multipliers of certain vector-valued reproducing kernel Hilbert spaces, linking their purity to the value at zero, and explores applications in multivariable function spaces.

Contribution

It establishes a new equivalence between the purity of multipliers and their value at zero for several important RKHSs, extending understanding of operator multipliers in multivariable settings.

Findings

Pure contractive multipliers are characterized by their value at zero.

The result applies to Hardy, Bergman, and Drury-Arveson spaces.

Applications to the polydisc are demonstrated.

Abstract

A contraction $T$ on a Hilbert space $\mathcal{H}$ is said to be pure if the sequence $\lbrace T^{*n} \rbrace_{n}$ 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 $X = (X_1,\ldots,X_n)$ on $\mathcal{H}$, the following statements are equivalent: (i) $T$ is a pure contraction on $\mathcal{H}$, (ii) the compression $P_{\mathcal{W}(X)}T|_{\mathcal{W}(X)}$ is a pure contraction, where $\mathcal{W}(X)$ is the wandering subspace corresponding to the tuple $X$. An operator-valued multiplier $\Phi$ of a vector-valued reproducing kernel Hilbert space (rkHs) is said to be pure contractive if the associated multiplication operator $M_{\Phi}$ is a pure contraction. Using the above result, we find that operator-valued mulitpliers $\Phi(\textbf{z})$ of several vector-valued rkHs's on the polydisc $\mathbb{D}^n$ as well as the unit ball $\mathbb{B}_n$ in $\mathbb{C}^n$ are pure contractive if and only if $\Phi(0)$ 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.