Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$k$-hyponormality of finite rank perturbations of unilateral weighted shifts


Authors: Raúl E. Curto and Woo Young Lee
Journal: Trans. Amer. Math. Soc. 357 (2005), 4719-4737
MSC (2000): Primary 47B20, 47B35, 47B37; Secondary 47-04, 47A20, 47A57
Published electronically: June 29, 2005
MathSciNet review: 2165385
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 47B20, 47B35, 47B37, 47-04, 47A20, 47A57

Retrieve articles in all journals with MSC (2000): 47B20, 47B35, 47B37, 47-04, 47A20, 47A57


Additional Information

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

Woo Young Lee
Affiliation: Department of Mathematics, Seoul National University, Seoul 151-742, Korea
Email: wylee@math.snu.ac.kr

DOI: http://dx.doi.org/10.1090/S0002-9947-05-04029-8
PII: S 0002-9947(05)04029-8
Keywords: Weighted shifts, perturbations, subnormal, $k$-hyponormal, weakly $k$-hyponormal
Received by editor(s): December 10, 1999
Received by editor(s) in revised form: December 31, 2001
Published electronically: June 29, 2005
Additional Notes: The work of the first-named author was partially supported by NSF research grants DMS-9800931 and DMS-0099357.
The work of the second-named author was partially supported by a grant (R14-2003-006-01001-0) from the Korea Science and Engineering Foundation.
Article copyright: © Copyright 2005 American Mathematical Society