Pure contractive multipliers of some reproducing kernel Hilbert spaces and applications

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.