k-hyponormality of finite rank perturbations of unilateral weighted shifts

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 $D_n(s):=\text{det} P_n [(W_\alpha+sW_\alpha^2)^*, W_\alpha+s W_\alpha^2] P_n$ 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.