Numerical ranges of weighted shifts

Let A be a unilateral (resp., bilateral) weighted shift with weights w n , 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 { a n } n = 0 ∞ (resp., { a n } n = − ∞ ∞ ) in [ − 1 , 1 ] such that | w n | 2 = ( 1 − a n ) ( 1 + a n + 1 ) for all n. In terms of such a n ʼs, we obtain a necessary and sufficient condition for W ( A ) to be open. If the w n ʼs are periodic, we show that the a n ʼs 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.