Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T09:03:38.193Z Has data issue: false hasContentIssue false
  • English
  • Français

L(n)-HYPONORMALITY: A MISSING BRIDGE BETWEEN SUBNORMALITY AND PARANORMALITY

Part of: Special classes of linear operators

Published online by Cambridge University Press:  01 April 2010

IL BONG JUNG ,
SUN HYUN PARK  and
JAN STOCHEL
IL BONG JUNG*
Affiliation:
Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea (email: [email protected])
SUN HYUN PARK
Affiliation:
Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea (email: [email protected])
JAN STOCHEL
Affiliation:
Instytut Matematyki, Uniwersytet Jagielloński, ul. Łojasiewicza 6, PL-30348 Kraków, Poland (email: [email protected])
*
✉For correspondence; e-mail: [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.

A new notion of L(n)-hyponormality is introduced in order to provide a bridge between subnormality and paranormality, two concepts which have received considerable attention from operator theorists since the 1950s. Criteria for L(n)-hyponormality are given. Relationships to other notions of hyponormality are discussed in the context of weighted shift and composition operators.

Keywords

subnormal operatorparanormal operatorL(n)-hyponormal operatorweighted shift operatorcomposition operator

MSC classification

Secondary: 47B20: Subnormal operators, hyponormal operators, etc. 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 88 , Issue 2 , April 2010 , pp. 193 - 203
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

The work of the first author was supported by a Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (No. R01-2008-000-20088-0). The work of the third author was supported by MNiSzW grant N201 026 32/1350.

References

[1]Agler, J., ‘The Arveson extension theorem and coanalytic models’, Integral Equations Operator Theory 5 (1982), 608–631.CrossRefGoogle Scholar
[2]Agler, J., ‘Hypercontractions and subnormality’, J. Operator Theory 13 (1985), 203–217.Google Scholar
[3]Ando, T., ‘Operators with a norm condition’, Acta Sci. Math. (Szeged) 33 (1972), 169–178.Google Scholar
[4]Athavale, A., ‘On joint hyponormality of operators’, Proc. Amer. Math. Soc. 103 (1988), 417–423.CrossRefGoogle Scholar
[5]Bram, J., ‘Subnormal operators’, Duke Math. J. 22 (1955), 75–94.CrossRefGoogle Scholar
[6]Burnap, C. and Jung, I., ‘Composition operators with weak hyponormality’, J. Math. Anal. Appl. 337 (2008), 686–694.CrossRefGoogle Scholar
[7]Chō, M. and Lee, J., ‘p-hyponormality is not translation-invariant’, Proc. Amer. Math. Soc. 131 (2003), 3109–3111.CrossRefGoogle Scholar
[8]Curto, R. E., ‘Quadratically hyponormal weighted shift’, Integral Equations Operator Theory 13 (1990), 49–66.CrossRefGoogle Scholar
[9]Curto, R. E., ‘Joint hyponormality: a bridge between hyponormality and subnormality’, Proc. Sym. Math. 51 (1990), 69–91.CrossRefGoogle Scholar
[10]Curto, R. E. and Lee, S. H., ‘Quartically hyponormal weighted shifts need not be 3-hyponormal’, J. Math. Anal. Appl. 314 (2006), 455–463.CrossRefGoogle Scholar
[11]Curto, R. E., Muhly, P. S. and Xia, J., ‘Hyponormal pairs of commuting operators’, Oper. Theory Adv. Appl. 35 (1988), 1–22.Google Scholar
[12]Embry, M., ‘A generalization of the Halmos–Bram condition for subnormality’, Acta. Sci. Math. (Szeged) 35 (1973), 61–64.Google Scholar
[13]Furuta, T., ‘On the class of paranormal operators’, Proc. Japan. Acad. 43 (1967), 594–598.Google Scholar
[14]Furuta, T., Invitation to Linear Operators (Taylor & Francis, London, 2001).CrossRefGoogle Scholar
[15]Furuta, T., Ito, M. and Yamazaki, T., ‘A subclass of paranormal operators including class of log-hyponormal and several related classes’, Sci. Math. 1 (1998), 389–403.Google Scholar
[16]Istrǎţescu, V., Saitô, T. and Yoshino, T., ‘On a class of operators’, Tôhoku Math. J. 18 (1966), 410–413.CrossRefGoogle Scholar
[17]Jung, I., Lee, M. and Park, S., ‘Separating classes of composition operators’, Proc. Amer. Math. Soc. 135 (2007), 3955–3965.CrossRefGoogle Scholar
[18]Jung, I. and Li, C., ‘A formula for k-hyponormality of backstep extensions of subnormal weighted shifts’, Proc. Amer. Math. Soc. 129 (2000), 2343–2351.CrossRefGoogle Scholar
[19]Halmos, P. R., ‘Normal dilations and extensions of operators’, Summa Bras. Math. 2 (1950), 124–134.Google Scholar
[20]Halmos, P. R., Measure Theory (van Nostrand, Princeton, NJ, 1956).Google Scholar
[21]Lambert, A., ‘Subnormality and weighted shifts’, J. London Math. Soc. 14 (1976), 476–480.CrossRefGoogle Scholar
[22]Lubin, A., ‘Weighted shifts and products of subnormal operators’, Indiana Univ. Math. J. 26 (1977), 839–845.CrossRefGoogle Scholar
[23]McCullough, S. and Paulsen, V. I., ‘A note on joint hyponormality’, Proc. Amer. Math. Soc. 107 (1989), 187–195.CrossRefGoogle Scholar
[24]McCullough, S. and Paulsen, V. I., ‘k-hyponormality of weighted shifts’, Proc. Amer. Math. Soc. 116 (1992), 165–169.Google Scholar
[25]Singh, R. K. and Manhas, J. S., Composition Operators on Function Spaces, North-Holland Mathematics Studies, 179 (North-Holland, Amsterdam, 1993).Google Scholar
[26]Šmul′jan, J. L., ‘An operator Hellinger integral’, Mat. Sb. 49 (1959), 381–430 (in Russian).Google Scholar
[27]Stochel, J. and Szafraniec, F. H., ‘On normal extensions of unbounded operators, II’, Acta Sci. Math. (Szeged) 53 (1989), 153–177.Google Scholar