標題: Numerical ranges of weighted shifts 作者: Wang, Kuo-ZhongWu, Pei Yuan應用數學系Department of Applied Mathematics 關鍵字: Numerical range;Numerical radius;Numerical contraction;Unilateral weighted shift;Bilateral weighted shift 公開日期: 15-九月-2011 摘要: 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.2011.04.010http://hdl.handle.net/11536/19226 ISSN: 0022-247X DOI: 10.1016/j.jmaa.2011.04.010 期刊: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS Volume: 381 Issue: 2 起始頁: 897 結束頁: 909 顯示於類別： 期刊論文