Title: Numerical ranges of weighted shifts
Authors: Wang, Kuo-Zhong
Wu, Pei Yuan
Department of Applied Mathematics
Keywords: Numerical range;Numerical radius;Numerical contraction;Unilateral weighted shift;Bilateral weighted shift
Issue Date: 15-Sep-2011
Abstract: Let A be a unilateral (resp., bilateral) weighted shift with weights w(n), n >= 0 (resp., -infinity < n < infinity). Eckstein and Racz showed before that A has its numerical range W (A) contained in the closed unit disc if and only if there is a sequence {a(n)}(n=0)(infinity) (resp., {a(n))(n=-infinity)(infinity)) in [-1,1] such that |w(n)|(2) = (1 - a(n))(1 + a(n+1)) for all n. In terms of such a(n)'s, we obtain a necessary and sufficient condition for W (A) to be open. If the w(n)'s are periodic, we show that the a(n)'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. (C) 2011 Elsevier Inc. All rights reserved.
URI: http://dx.doi.org/10.1016/j.jmaa.2011.04.010
ISSN: 0022-247X
DOI: 10.1016/j.jmaa.2011.04.010
Volume: 381
Issue: 2
Begin Page: 897
End Page: 909
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