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Separating classes of composition operators via subnormal condition


Authors: Il Bong Jung, Mi Ryeong Lee and Sang Soo Park
Journal: Proc. Amer. Math. Soc. 135 (2007), 3955-3965
MSC (2000): Primary 47B20, 47B33; Secondary 47A63
DOI: https://doi.org/10.1090/S0002-9939-07-09003-X
Published electronically: June 19, 2007
MathSciNet review: 2341946
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Abstract: Several classes have been considered to study the weak subnormalities of Hilbert space operators. One of them is $ n$-hypnormality, which comes from the Bram-Halmos criterion for subnormal operators. In this note we consider $ E(n)$-hyponormality, which is the parallel version corresponding to the Embry characterization for subnormal operators. We characterize $ E(n)$ -hyponormality of composition operators via $ k$-th Radon-Nikodym derivatives and present some examples to distinguish the classes.


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

  • [1] J. Agler, Hypercontractions and subnormality, J. Operator Theory, 13(1985), 203-217. MR 775993 (86i:47028)
  • [2] C. Burnap and I. Jung, Composition operators with weak hyponormality, J. Math. Anal. Appl., to appear.
  • [3] C. Burnap, I. Jung and A. Lambert, Separating partial normality classes with composition operators, J. Operator Theory, 53(2005), 381-397. MR 2153155
  • [4] R. Curto, Quadratically hyponormal weighted shifts, Integral Equation Operator Theory 13(1990), 49-66. MR 1025673 (90k:47061)
  • [5] -, Joint hyponormality: A bridge between hyponormality and subnormality, Proc. Sympos. Math. 51(1990), 69-91. MR 1025673 (90k:47061)
  • [6] R. Curto and L. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, Integral Equations Operator Theory, 17(1993), 202-246. MR 1233668 (94h:47050)
  • [7] -, Recursively generated weighted shifts and the subnormal completion problem II, Integral Equations Operator Theory, 18(1994), 369-426. MR 1233668 (94h:47050)
  • [8] -, Solution of the truncated complex moment problems for flat data, Memoirs Amer. Math. Soc. 568(1996). MR 1233668 (94h:47050)
  • [9] R. Curto, S. Lee and J. Yoon, k-Hyponormality of multivariable weighted shifts, J. Funct. Anal., 229(2005), 462-480. MR 2183156
  • [10] R. Curto and W. Lee, Joint hyponormality of Toeplitz pairs, Memoirs of Amer. Math. Soc., Vol. 150, No. 712 (2001). MR 1810770 (2002c:47042)
  • [11] M. Embry, A generalization of the Halmos-Bram condition for subnormality, Acta. Sci. Math.(Szeged) 35 (1973), 61-64. MR 0328652 (48:6994)
  • [12] M. Embry and A. Lambert, Subnormality for the adjoint of a composition operator on $ L^{2}$, J. Operator Theory, 25 (1991), 309-318. MR 1203036 (94f:47028)
  • [13] G. Exner, On $ n$-contractive and $ n$- hypercontractive operators, Integral Equations Operator Theory, 56 (2006), 451-468.
  • [14] G. Exner, I. Jung, and S. Park, On $ n$ -hypercontractive operators, II, submitted.
  • [15] T. Furuta, Invitation to linear operators, Taylor & Francis Inc., 2001. MR 1978629 (2004b:47001)
  • [16] M. Ito and T. Yamazaki, Relations between two inequalities $ (B^{r/2}A^{p}B^{r/2})^{r/(p+r)}\geq B^{r}$ and $ A^{p}\geq (A^{p/2}B^{r}A^{p/2})^{p/(p+r)}$ and their applications, Integral Equations Operator Theory, 44(2002), 442-450. MR 1942034 (2003h:47032)
  • [17] I. Jung, E. Ko, C. Li and S. Park, Embry truncated complex moment problem, Linear Algebra and Appl. 375 (2003), 95-114. MR 2013458 (2004i:47030)
  • [18] I. Jung, C. Li and S. Park, Complex moment matrices via Halmos-Bram and Embry conditions, J. Korean Math. Soc., to appear.
  • [19] I. Jung and C. Li, A formula for $ k$ -hyponormality of backstep extensions of subnormal weighted shifts, Proc. Amer. Math. Soc. 129(2000), 2243-2351. MR 1823917 (2002b:47061)
  • [20] A. Lambert, Hyponormal composition operators, Bull. London Math. Soc. 18(1986), 395-400. MR 838810 (87h:47059)
  • [21] S. McCullough and V. I. Paulsen, A note on joint hyponormality, Proc. Amer. Math. Soc. 107(1989), 187-195. MR 972236 (90a:47062)
  • [22] , $ k$ -hyponormality of weighted shifts, Proc. Amer. Math. Soc. 116 (1992), 165-169. MR 1102858 (93e:47029)
  • [23] M. Rao, Conditional measures and applications, Marcel Dekker, New York, 1993. MR 1234936 (95d:28001)
  • [24] J. Park and S. Park, On $ k$-hyponormal weighted translation semigroups, Bull. Kor. Math. Soc. 39(2002), No. 4, 527-534. MR 1938992 (2003h:47042)
  • [25] J. Shohat and J. Tamarkin, The problem of moments, Math. Surveys I, Amer. Math. Soc., Providence, 1943. MR 0008438 (5:5c)
  • [26] R. Singh and J. Manhas, Composition operators on function spaces, North-Holland Math. Stud. No. 179, Amsterdam, 1993. MR 1246562 (95d:47036)

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Additional Information

Il Bong Jung
Affiliation: Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu 702-702 Korea
Email: ibjung@knu.ac.kr

Mi Ryeong Lee
Affiliation: Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu 702-702 Korea
Email: lmr67@yumail.ac.kr

Sang Soo Park
Affiliation: Institute of Mathematical Science, Ewha Womans University, Seoul, 120-750, Korea
Email: pss4855@ewha.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-07-09003-X
Keywords: Composition operator, subnormal operator.
Received by editor(s): June 14, 2006
Received by editor(s) in revised form: November 7, 2006
Published electronically: June 19, 2007
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2007 American Mathematical Society

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