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

2020/09/14 by Raúl E. Curto, Sang Hoon Lee, Curto, Raul E. +3
Mathematics · #47 #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2009.06130

openalex publication_date 2020/09/14 · openalex created_date 2020/09/21 · openalex updated_date 2026/07/28

Abstract

Given a bounded sequence ωof positive numbers and its associated unilateral weighted shift W_ω acting on the Hilbert space ℓ²(ℤ₊), we consider natural representations of W_ω as a 2-variable weighted shift, acting on ℓ²(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 \∫ p(r)^k q(r)^ℓ dσ(r)\_{k,ℓ ∈ ℤ+}, where W_ω is assumed to be subnormal with Berger measure σsuch that \supp σ⊆ 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 A₂ 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≥ 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.

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