k-hyponormality of finite rank perturbations of unilateral weighted shifts

Abstract

In this paper we explore finite rank perturbations of unilateral weighted shifts $W_{\alpha }$. First, we prove that the subnormality of $W_{\alpha }$ 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 of $D_n(s):=\text{det}{P}_n[(W_{\alpha }+s{W}_{\alpha }^2{)}^{\ast },W_{\alpha }+s{W}_{\alpha }^2]P_n$ are nonnegative, for every $n\ge 0$, where $P_n$ denotes the orthogonal projection onto the basis vectors $\{e_0,...,e_n\}$. Finally, for $\alpha$ strictly increasing and $W_{\alpha }$ 2-hyponormal, we show that for a small finite-rank perturbation ${\alpha }^{\prime }$ of $\alpha$, the shift $W_{{\alpha }^{\prime }}$ remains quadratically hyponormal.

19 pages; to appear in Trans. Amer. Math. Soc