A distinction of k-hyponormal and weakly k-hyponormal weighted shifts

Abstract

Let ?(x) : ?x/2, ?2/3, ?3/4, ?4/5, ... be a sequence with a real variable x > 0 and let W?(x) be the associated weighted shift with weight sequence ?(x). In [17], Exner-Jung-Park provided an algorithm to distinguish weak k-hyponormality and k-hyponormality of weighted shift W?(x), and obtained sn > 0 for some low numbers n = 4,..., 10, such that W?(sn) is weakly n-hyponormal but not n-hyponormal. In this paper, we obtain a formula of sn (for all positive integer n) such that W?(sn) is weakly n-hyponormal but not n-hyponormal, which improves Exner-Jung-Park?s result above.