Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-30T20:14:06.276Z Has data issue: false hasContentIssue false
  • English
  • Français

BACKWARD 3-STEP EXTENSIONS OF RECURSIVELY GENERATED WEIGHTED SHIFTS: A RANGE OF QUADRATIC HYPONORMALITY

Part of: Special classes of linear operators

Published online by Cambridge University Press:  20 November 2013

GEORGE R. EXNER ,
IL BONG JUNG ,
MI RYEONG LEE  and
SUN HYUN PARK
GEORGE R. EXNER
Affiliation:
Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837, USA email [email protected]
IL BONG JUNG*
Affiliation:
Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea email [email protected]
MI RYEONG LEE
Affiliation:
Institute of Liberal Education, Catholic University of Daegu, Gyeongsan 712-702, Korea email [email protected]
SUN HYUN PARK
Affiliation:
Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $\alpha : 1, 1, \sqrt{x} , \mathop{( \sqrt{u} , \sqrt{v} , \sqrt{w} )}\nolimits ^{\wedge } $ be a backward 3-step extension of a recursively generated weighted sequence of positive real numbers with $1\leq x\leq u\leq v\leq w$ and let ${W}_{\alpha } $ be the associated weighted shift with weight sequence $\alpha $. The set of positive real numbers $x$ such that ${W}_{\alpha } $ is quadratically hyponormal for some $u, v$ and $w$ is described, solving an open problem due to Curto and Jung [‘Quadratically hyponormal weighted shifts with two equal weights’, Integr. Equ. Oper. Theory 37 (2000), 208–231].

Keywords

weighted shiftsquadratic hyponormalitypositive quadratic hyponormalitysubnormal completion

MSC classification

Secondary: 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47B20: Subnormal operators, hyponormal operators, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 3 , June 2014 , pp. 488 - 493
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Bae, J. Y., Exner, G. and Jung, I. B., ‘Criteria for positively quadratically hyponormal weighted shifts’, Proc. Amer. Math. Soc. 130 (2002), 3287–3294.CrossRefGoogle Scholar
Curto, R., ‘Joint hyponormality: a bridge between hyponormality and subnormality’, Proc. Sympos. Pure Math. 51 (1990), 69–91.CrossRefGoogle Scholar
Curto, R., ‘Quadratically hyponormal weighted shifts’, Integr. Equ. Oper. Theory 13 (1990), 49–66.CrossRefGoogle Scholar
Curto, R., ‘Polynomially hyponormal operators on Hilbert space’, Proc. ELAM VII Revista Union Mat. Arg. 37 (1991), 29–56.Google Scholar
Curto, R. and Fialkow, L., ‘Recursively generated weighted shifts and the subnormal completion problem, II’, Integr. Equ. Oper. Theory 18 (1994), 369–426.Google Scholar
Curto, R. and Jung, I. B., ‘Quadratically hyponormal weighted shifts with two equal weights’, Integr. Equ. Oper. Theory 37 (2000), 208–231.Google Scholar
Curto, R. and Lee, W. Y., ‘Solution of the quadratically hyponormal completion problem’, Proc. Amer. Math. Soc. 131 (2003), 2479–2489.Google Scholar
Exner, G., Jung, I. B., Lee, M. R. and Park, S. H., ‘Quadratically hyponormal weighted shifts with recursive tail’, J. Math. Anal. Appl. 408 (2013), 298–305.Google Scholar
Exner, G., Jung, I. B. and Park, D. W., ‘Some quadratically hyponormal weighted shifts’, Integr. Equ. Oper. Theory 60 (2008), 13–36.CrossRefGoogle Scholar
Jung, I. B. and Park, S. S., ‘Quadratically hyponormal weighted shifts and their examples’, Integr. Equ. Oper. Theory 36 (2002), 480–498.Google Scholar
Poon, Y. T. and Yoon, J., ‘Quadratically hyponormal recursively generated weighted shifts need not be positively quadratically hyponormal’, Integr. Equ. Oper. Theory 58 (2007), 551–562.Google Scholar
Stampfli, J., ‘Which weighted shifts are subnormal’, Pacific J. Math. 17 (1966), 367–379.CrossRefGoogle Scholar