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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Extensions and extremality of recursively generated weighted shifts


Authors: Raúl E. Curto, Il Bong Jung and Woo Young Lee
Journal: Proc. Amer. Math. Soc. 130 (2002), 565-576
MSC (1991): Primary 47B20, 47B37; Secondary 47-04, 47A57, 15A57
Published electronically: June 22, 2001
MathSciNet review: 1862138
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Abstract: 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

\begin{displaymath}W_{\alpha }\text{ is subnormal } \Longleftrightarrow \ \begin... ...text{ is $([\frac{k+1}{2}]+2)$ -hyponormal} & (n>1).\end{cases}\end{displaymath}

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.


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Additional Information

Raúl E. Curto
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
Email: curto@math.uiowa.edu

Il Bong Jung
Affiliation: Department of Mathematics, Kyungpook National University, Taegu 702–701, Korea
Email: ibjung@bh.kyungpook.ac.kr

Woo Young Lee
Affiliation: Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea
Email: wylee@yurim.skku.ac.kr

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06079-8
PII: S 0002-9939(01)06079-8
Keywords: Extensions of weighted shifts, recursively generated shifts, $k$-hyponormality
Received by editor(s): July 14, 2000
Published electronically: June 22, 2001
Additional Notes: The work of the first-named author was partially supported by NSF research grants DMS-9401455 and DMS-9800931.
The work of the second-named author was partially supported by KOSEF, research grant 2000-1-10100-002-3
The work of the third-named author was partially supported by the Brain Korea 21 Project.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2001 American Mathematical Society