Numerical ranges of weighted shifts

作者：Wang Kuo Zhong^*; Wu Pei Yuan

来源：Journal of Mathematical Analysis and Applications, 2011, 381(2): 897-909.

DOI：10.1016/j.jmaa.2011.04.010

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

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.

• 出版日期2011-9-15