k-HYPONORMALITY OF FINITE RANK PERTURBATIONS OF UNILATERAL WEIGHTED SHIFTS

Abstract. In this paper we explore finite rank perturbations of unilateral weighted shifts Wα. First, we prove that the subnormality of Wα is never stable under nonzero finite rank pertrubations 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 ofDn(s) := detPn [(Wα+sW 2 α) , Wα+sW 2 α]Pn are nonnegative, for every n ≥ 0, where Pn denotes the orthogonal projection onto the basis vectors {e0, · · · , en}. Finally, for α strictly increasing and Wα 2-hyponormal, we show that for a small finite-rank perturbation α of α, the shift W α ′ remains quadratically hyponormal.