Journal article
Polynomial Embeddings of Unilateral Weighted shifts in 2–Variable Weighted Shifts
Integral equations and operator theory, Vol.93(6), 64
12/02/2021
DOI: 10.1007/s00020-021-02681-1
Abstract
Given a bounded sequence ω of positive numbers, and its associated unilateral weighted shift Wω acting on the Hilbert space ℓ2(Z+), we consider natural representations of Wω as a 2–variable weighted shift, acting on the Hilbert space ℓ2(Z2+). Alternatively, we seek to examine the various ways in which the sequence ω 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⊆R+ and the double-indexed moment sequence {∫p(r)kq(r)ℓdσ(r)}k,ℓ∈Z+, where Wω is assumed to be subnormal with Berger measure σ such that suppσ⊆I; we call such an embedding a (p, q)–embedding of Wω. We prove that every (p, q)–embedding of a subnormal weighted shift Wω 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 (jointly) hyponormal 2–variable weighted shift Θ(Wω) (where Θ(Wω) denotes the classical embedding of a hyponormal unilateral weighted shift Wω) such that some integer power of Θ(Wω) is not hyponormal? As another application, we find an alternative way to compute the Berger measure of the Agler j–th shift Aj (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.
Details
- Title: Subtitle
- Polynomial Embeddings of Unilateral Weighted shifts in 2–Variable Weighted Shifts
- Creators
- Raúl E Curto - University of IowaSang Hoon Lee - Chungnam National UniversityJasang Yoon - The University of Texas Rio Grande Valley
- Resource Type
- Journal article
- Publication Details
- Integral equations and operator theory, Vol.93(6), 64
- Publisher
- Springer International Publishing
- DOI
- 10.1007/s00020-021-02681-1
- ISSN
- 0378-620X
- eISSN
- 1420-8989
- Grant note
- University of Texas System CONACYT Mexico 2020R1A2C1A0100584611 / National Research Foundation of Korea (http://dx.doi.org/10.13039/501100003725)
- Language
- English
- Date published
- 12/02/2021
- Academic Unit
- Mathematics
- Record Identifier
- 9984241055202771
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