A distinction of $k$-hyponormal and weakly $k$-hyponormal weighted shifts

Abstract

Let $\alpha (x):\sqrt{x},\sqrt{\frac{2}{3}},\sqrt{\frac{3}{4}},\sqrt{\frac{4}{5}},...$ be a sequence with a real variable $x>0$ and let $W_{\alpha\left(x\right) }$ be the associated weighted shift with weight sequence $\alpha (x)$. In \cite{EJP}, Exner-Jung-Park provided an algorithm to distinguish weak $k$-hyponormality and $k$-hyponormality of weighted shift $W_{\alpha \left( x\right) }$, and obtained $s_{n}>0$ for some low numbers $n=4,...,10$, such that $W_{\alpha \left( s_{n}\right) }$ is weakly $n$-hyponormal but not $n$-hyponormal. In this paper, we obtain a formula of $s_{n}$ (for all positive integer $n$) such that $W_{\alpha \left(s_{n}\right)}$ is weakly $n$-hyponormal but not $n$-hyponormal, which improves Exner-Jung-Park's result above.