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A contraction $T$ on a Hilbert space $\mathcal{H}$ is said to be pure if thesequence $\lbrace T^{*n} \rbrace_{n}$ converges to $0$ in the strong operatortopology. In this article, we prove that for contractions $T$, which commutewith certain tractable tuples of commuting operators $X = (X_1,\ldots,X_n)$ on$\mathcal{H}$, the following statements are equivalent: (i) $T$ is a purecontraction 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 $X$. Anoperator-valued multiplier $\Phi$ of a vector-valued reproducing kernel Hilbertspace (rkHs) is said to be pure contractive if the associated multiplicationoperator $M_{\Phi}$ is a pure contraction. Using the above result, we find thatoperator-valued mulitpliers $\Phi(\textbf{z})$ of several vector-valued rkHs'son 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 purecontraction on the underlying Hilbert space. The list includes Hardy, Bergmanand Drury-Arveson spaces. Finally, we present some applications of ourcharacterization of pure contractive multipliers associated with the polydisc.

Pure contractive multipliers of some reproducing kernel Hilbert spaces and applications