A note on coherent states related to weighted shifts
type: Item
instance of: scholarly article
title: A note on coherent states related to weighted shifts
zbMATH Open document ID: 0814.47034
author: Włodzimierz Mlak
author: Jan Stochel
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1995-06-12T00:00:00Z
review text: If \(W\) is a weighted shift with positive weights and the coherence radius of \(W\), \(r(W)> 0\), then the authors show that \(h(z)\in D(W^*)\), the domain of the adjoint \(W^*\) of \(W\), where \(h(z)= e_ 0+ \sum^ \infty_{n= 1} {z^ n\over w_ 0\cdot\cdots\cdot w_{n-1}} e_ n\) with \(e_ n(z)= {z^ n\over \sqrt{n!}}\) for \(n\geq 0\), \[ z\in S(W)= \{z\in \mathbb{C}: | z|< r(W)\}. \] Also \(W^* h(z)= zh(z)\), \(z\in S(W)\) and \(\dim\text{Ker}(zI- W^*)= 1\), \(z\in S(W)\). Next, if \(W\) is a subnormal weighted shift with positive weights and if \(\mu(\mathbb{C}- S(W))= 0\), then the map \(f\to \widehat f\) is a unitary isomorphism of a separable Hilbert space \(H\) onto the set \(B^ 2(\mu)\) of all entire functions belonging to \(L^ 2(\mu)\). Furthermore, the identity operator \(I\) on \(H\) is given by \(I= \int_{S(W)} P_ z\| h(\bar z)\|^ 2 d\mu(z)\). Here, given \(z\in S(W)\), \(P_ z\) denotes the orthogonal projection of \(H\) onto the one-dimensional space spanned by the vector \(h(z)\).
Mathematics Subject Classification ID: 47B37
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 47B20
Mathematics Subject Classification ID: 81R30
zbMATH DE Number: 597457
zbMATH Keywords: coherence radius
zbMATH Keywords: subnormal weighted shift with positive weights
zbMATH Keywords: unitary isomorphism
zbMATH Keywords: orthogonal projection
reviewed by: K. Chandrasekhara Rao
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195168208
OpenAlex ID: W2048922557
cites work: On a Hilbert space of analytic functions and an associated integral transform part I
cites work: Q3337807
cites work: Q4059478
cites work: On normal extensions of unbounded operators. III: Spectral properties
DOI: 10.2977/PRIMS/1195168208
description: scientific article; zbMATH DE number 597457