Paper detail

Polynomial embeddings of unilateral weighted shifts into $2$-variable weighted shifts

Given a bounded sequence ωof positive numbers and its associated unilateral weighted shift W_ω acting on the Hilbert space \ell^2(\mathbb{Z}_+), we consider natural representations of W_ω as a 2-variable weighted shift, acting on \ell^2(\mathbb{Z}_+^2). Alternatively, we seek to examine the various ways in which the sequence ωcan give rise to a 2-variable weight diagram. Our best (and more general) embedding arises from looking at two polynomials p and q nonnegative on a closed interval I in R_+ and the double-indexed moment sequence \{\int p(r)^k q(r)^{\ell} dσ(r)\}_{k,\ell \in \mathbb{Z}_+}, where W_ω is assumed to be subnormal with Berger measure σsuch that \supp \; σ\subseteq I; we call such an embedding a (p,q)-embedding of W_ω. We prove that every (p,q)-embedding of a subnormal weighted shift W_ω 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_{(α,β)}? If so, what is the Berger measure of W_{(α,β)}? (ii) Can a contractive subnormal unilateral weighted shift be always embedded in a spherically isometric 2-variable weighted shift? (iii) Does there exist a hyponormal 2-variable weighted shift Θ(W_ω) (where Θ(W_ω) denotes the classical embedding of a hyponormal unilateral weighted shift W_ω) such that some integer power of Θ(W_ω) 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\geq 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.