$k$-hyponormality of finite rank perturbations of unilateral weighted shifts

Abstract

In this paper we explore finite rank perturbations of unilateral weighted shifts W α . First, we prove that the subnormality of W α is never stable under nonzero finite rank perturbations unless the perturbation occurs at the zeroth weight. Second, we establish that 2-hyponormality implies positive quadratic hyponormality, in the sense that the Maclaurin coefficients of D n (s):= det P n [(W α + sW 2 α )*, W α + sW 2 α ]P n are nonnegative, for every n > 0, where P n denotes the orthogonal projection onto the basis vectors {e 0 , ..., e n }. Finally, for α strictly increasing and W α 2-hyponormal, we show that for a small finite-rank perturbation a' of a, the shift W α , remains quadratically hyponormal.