Numerical ranges of weighted shifts

Kuo-Zhong Wang*, Pei Yuan Wu

*Corresponding author for this work

Research output: Contribution to journalArticle

12 Scopus citations

Abstract

Let A be a unilateral (resp., bilateral) weighted shift with weights wn, n≤0 (resp., -∞<n<∞). Eckstein and Rácz showed before that A has its numerical range W(A) contained in the closed unit disc if and only if there is a sequence {an}n=0 (resp., {an}n=0) in [-1,1] such that |wn|2=(1-an)(1+an+1) for all n. In terms of such an's, we obtain a necessary and sufficient condition for W(A) to be open. If the wn's are periodic, we show that the an's can also be chosen to be periodic. As a result, we give an alternative proof for the openness of W(A) for an A with periodic weights, which was first proven by Stout. More generally, a conjecture of his on the openness of W(A) for A with split periodic weights is also confirmed.

Original languageEnglish
Pages (from-to)897-909
Number of pages13
JournalJournal of Mathematical Analysis and Applications
Volume381
Issue number2
DOIs
StatePublished - 15 Sep 2011

Keywords

  • Bilateral weighted shift
  • Numerical contraction
  • Numerical radius
  • Numerical range
  • Unilateral weighted shift

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