Polynomial Embeddings of Unilateral Weighted shifts in 2–Variable Weighted Shifts

Abstract

Given a bounded sequence $$\omega $$ of positive numbers, and its associated unilateral weighted shift $$W_{\omega }$$ acting on the Hilbert space $$\ell ^2(\mathbb {Z}_+)$$ , we consider natural representations of $$W_{\omega }$$ as a 2–variable weighted shift, acting on the Hilbert space $$\ell ^2(\mathbb {Z}_+^2)$$ . Alternatively, we seek to examine the various ways in which the sequence $$\omega $$ can give rise to a 2–variable weight diagram, corresponding to a 2–variable weighted shift. Our best (and more general) embedding arises from looking at two polynomials p and q nonnegative on a closed interval $$I \subseteq \mathbb {R}_+$$ and the double-indexed moment sequence $$\{\int p(r)^k q(r)^{\ell } d\sigma (r)\}_{k,\ell \in \mathbb {Z}_+}$$ , where $$W_{\omega }$$ is assumed to be subnormal with Berger measure $$\sigma $$ such that $${\text {supp}}\; \sigma \subseteq I$$ ; we call such an embedding a (p, q)–embedding of $$W_{\omega }$$ . We prove that every (p, q)–embedding of a subnormal weighted shift $$W_{\omega }$$ is (jointly) subnormal, and we explicitly compute its Berger measure. We apply this result to answer three outstanding questions: (i) Can the Bergman shift $$A_2$$ be embedded in a subnormal 2–variable spherically isometric weighted shift $$W_{(\alpha ,\beta )}$$ ? If so, what is the Berger measure of $$W_{(\alpha ,\beta )}$$ ? (ii) Can a contractive subnormal unilateral weighted shift be always embedded in a spherically isometric 2–variable weighted shift? (iii) Does there exist a (jointly) hyponormal 2–variable weighted shift $$\Theta (W_{\omega })$$ (where $$\Theta (W_{\omega })$$ denotes the classical embedding of a hyponormal unilateral weighted shift $$W_{\omega }$$ ) such that some integer power of $$\Theta (W_{\omega })$$ is not hyponormal? As another application, we find an alternative way to compute the Berger measure of the Agler j–th shift $$A_{j}$$ ( $$j\ge 2$$ ). Our research uses techniques from the theory of disintegration of measures, Riesz functionals, and the functional calculus for the columns of the moment matrix associated to a polynomial embedding.