$k$-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 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 $D_{n}(s):=\text{det}\,P_{n}\,[(W_{\alpha }+sW_{\alpha }^{2})^{*},\, W_{\alpha }+s W_{\alpha }^{2}]\,P_{n}$are nonnegative, for every $n\ge 0$, where $P_{n}$ denotes the orthogonal projection onto the basis vectors $\{e_{0},\cdots ,e_{n}\}$. 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.