Extensions and extremality of recursively generated weighted shifts

Given an $n$-step extension $\alpha :x_{n},\cdots ,x_{1},(\alpha _{0},\cdots ,\alpha _{k})^{\wedge }$ of a recursively generated weight sequence $(0<\alpha _{0}<\cdots <\alpha _{k})$, and if $W_{\alpha }$ denotes the associated unilateral weighted shift, we prove that

In particular, the subnormality of an extension of a recursively generated weighted shift is independent of its length if the length is bigger than 1. As a consequence we see that if $\alpha (x)$ is a canonical rank-one perturbation of the recursive weight sequence $\alpha$, then subnormality and $k$-hyponormality for $W_{\alpha (x)}$ eventually coincide. We then examine a converse--an ``extremality" problem: Let $\alpha (x)$ be a canonical rank-one perturbation of a weight sequence $\alpha$ and assume that $(k+1)$-hyponormality and $k$-hyponormality for $W_{\alpha (x)}$ coincide. We show that $\alpha (x)$ is recursively generated, i.e., $W_{\alpha (x)}$is recursive subnormal.