论文标题
单方面加权转换的多项式嵌入到$ 2 $可变的加权转移
Polynomial embeddings of unilateral weighted shifts into $2$-variable weighted shifts
论文作者
论文摘要
给定一个有界的序列ωof正数及其相关的单侧加权移位W_Ω作用在Hilbert Space \ Ell^2（\ Mathbb {Z} _+）上，我们将W_Ω的自然表示为2个可加权移动的自然表示，作用于\ ell^2（\ ell^2）另外，我们试图检查序列ωcan产生2个可变的重量图的各种方式。我们最好的（并且更一般）嵌入是由于在r_+中查看两个多项式p和q nonngative以及双索引矩序\ {\ int p（r）^k q（r）^k q（r）^q（r）^{\ ell} d =（r）用Berger Measureσsuchus n \ supp \; σ\ subseteq i;我们称之为w_Ω的嵌入A（P，Q）。我们证明，加权偏移的每个（p，q）升压W_Ω是（共同）下正常的，并且我们明确计算其Berger度量。我们将此结果应用于回答以下三个突出的问题：（i）伯格曼移位A_2可以嵌入2个正常的2-变量球形等距加权移位中w _ {（α，β）}吗？如果是这样，W _ {（α，β）}的Berger量度是什么？ （ii）是否可以将收缩的单侧加权转移始终嵌入球形等距2变量的加权移位中？ （iii）是否存在不正常的2个加权移位θ（w_Ω）（其中θ（w_Ω）表示不良单侧加权移位w_Ω的经典嵌入，使得某些θ（w_Ω）的整数功率不是高度不良吗？作为另一个应用程序，我们找到了一种替代方法来计算Agler J-th Shift A_ {J}（J \ GEQ 2）的Berger度量。我们的研究使用了与多项式嵌入相关的矩矩阵的列的量度，Riesz功能分解理论和功能演算的技术。
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.
