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.