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Numerical ranges of weighted shifts

Numerical ranges of weighted shifts,10.1016/j.jmaa.2011.04.010,Journal of Mathematical Analysis and Applications,Kuo-Zhong Wang,Pei Yuan Wu

Numerical ranges of weighted shifts   (Citations: 2)
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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=−∞∞) in [−1,1] such that |wn|2=(1−an)(1+an+1) for all n. In terms of such anʼs, we obtain a necessary and sufficient condition for W(A) to be open. If the wnʼs are periodic, we show that the anʼs 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.
Journal: Journal of Mathematical Analysis and Applications - J MATH ANAL APPL , vol. 381, no. 2, pp. 897-909, 2011
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