On hyponormal weighted shift
type: Item
instance of: scholarly article
title: On hyponormal weighted shift
zbMATH Open document ID: 0542.47019
published in: Chinese Annals of Mathematics. Series B
publication date: 1984-01-01T00:00:00Z
review text: It is shown in this paper that a necessary condition for a hyponormal weighted unilateral shift of norm one to be unitarily equivalent to a Toeplitz operator is that the associated weights \(\{a_ n\}_ 0^{\infty}\) must satisfy \(1-| a_ n|^ 2=(1-| a_ 0|^ 2)^{n+1},\quad \forall n\geq 0.\) As a direct consequence, the answer to a problem of \textit{M. B. Abrahamse} [Duke Math. J. 43, 597-604 (1976; Zbl 0332.47017)] is shown, i.e. the Bergman shift is not unitarily equivalent to a Toeplitz operator. Recently, C. C. Cowen, J. Long and Sun Shunhua have independently shown that the condition \(1-| a_ n|^ 2=(1-| a_ 0|^ 2)^{n+1},\quad \forall n\geq 0\) is also sufficient for a hyponormal weighted shift of norm one to be unitarily equivalent to a Toeplitz operator. Also, S. Power has shown that the sequence \(\{a_ n\}_ 0^{\infty}\) satisfying \((1-| a_ n|^ 2)=(1-| a_ 0|^ 2)^{n+1},\quad \forall n\geq 0\) is a subnormal sequence. All these lead to a surprising answer to \textit{P. R. Halmos'}, 5th question [Bull. Am. Math. Soc. 76, 887-933 (1970; Zbl 0204.150)], i.e. there exists a subnormal Toeplitz operator neither normal nor analytic.
Mathematics Subject Classification ID: 47B20
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47B37
zbMATH DE Number: 3861847
zbMATH Keywords: hyponormal weighted unilateral shift
zbMATH Keywords: the Bergman shift is not unitarily equivalent to a Toeplitz operator
zbMATH Keywords: subnormal sequence
zbMATH Keywords: there exists a subnormal Toeplitz operator neither normal nor analytic
author: Shun Hua Sun
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3861847