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

Authors

Raul E. Curto
Sang H. Lee
Jasang Yoon, The University of Texas Rio Grande ValleyFollow

Document Type

Article

Publication Date

12-2021

Abstract

Given a bounded sequence omega of positive numbers, and its associated unilateral weighted shift W-omega, acting on the Hilbert space l(2) (Z+), we consider natural representations of W-omega as a 2-variable weighted shift, acting on the Hilbert space l(2)(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 subset of R+ and the double-indexed moment sequence {integral p(r)(k) q(r)(l) d sigma(r)}(k,l is an element of Z+) where W-omega is assumed to be subnormal with Berger measure a such that supp sigma subset of 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,W-beta)? If so, what is the Berger measure of W-(alpha,W-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 >= 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.

Recommended Citation

Curto, Raúl E., Sang Hoon Lee, and Jasang Yoon. "Polynomial Embeddings of Unilateral Weighted shifts in 2–Variable Weighted Shifts." Integral Equations and Operator Theory 93 (2021): 1-29. https://doi.org/10.1007/s00020-021-02681-1

Publication Title

Integral Equations and Operator Theory

DOI

10.1007/s00020-021-02681-1