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$arenonnegative,forevery$n\ge 0$,where$P_n$ denotes the orthogonal projection onto the basis vectors $\{e_0,\mathrm{.}\mathrm{.}\mathrm{.},e_n\}$. Finally, for $\alpha$ strictly increasing and $W_{\alpha }$ 2-hyponormal, we show that for a small finite-rank perturbation ${\alpha }^{\mathrm{\prime }}$ of $\alpha$, the shift $W_{{\alpha }^{\mathrm{\prime }}}$ remains quadratically hyponormal.