A formula for 𝑘-hyponormality of backstep extensions of subnormal weighted shifts

Jung, Il; Li, Chunji

doi:10.1090/S0002-9939-00-05844-5

A formula for $k$-hyponormality of backstep extensions of subnormal weighted shifts
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by Il Bong Jung and Chunji Li PDF

Proc. Amer. Math. Soc. 129 (2001), 2343-2351 Request permission

Abstract:

Let ${\alpha }: {\alpha }_{0}, {\alpha }_{1}, \cdots$ be a weight sequence of positive real numbers and let $W_{{\alpha }}$ be a subnormal weighted shift with a weight sequence $\alpha$. Consider an extended weight sequence ${\alpha }(x) : x, {\alpha }_{0}, {\alpha }_{1}, \cdots$ with $0<x \le \alpha _{0}$ and let $HE({\alpha ,k}):= \{ x >0: W_{\alpha (x)}\ \text {is} \ k \text {-hyponormal}\}$ for $k \in {\mathbb {N}}\cup \{\infty \}$, where $\mathbb {N}$ is the set of natural numbers. We obtain a formula to find the interval $HE({\alpha ,k})\setminus HE({\alpha ,k+1})$, which provides several examples to distinguish the classes of $k$-hyponormal operators from one another.

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