Separating classes of composition operators via subnormal condition

Jung, Il; Lee, Mi; Park, Sang

doi:10.1090/S0002-9939-07-09003-X

Separating classes of composition operators via subnormal condition
HTML articles powered by AMS MathViewer

by Il Bong Jung, Mi Ryeong Lee and Sang Soo Park PDF

Proc. Amer. Math. Soc. 135 (2007), 3955-3965 Request permission

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

Jim Agler, Hypercontractions and subnormality, J. Operator Theory 13 (1985), no. 2, 203–217. MR 775993
C. Burnap and I. Jung, Composition operators with weak hyponormality, J. Math. Anal. Appl., to appear.
Charles Burnap, Il Bong Jung, and Alan Lambert, Separating partial normality classes with composition operators, J. Operator Theory 53 (2005), no. 2, 381–397. MR 2153155
Raúl E. Curto, Quadratically hyponormal weighted shifts, Integral Equations Operator Theory 13 (1990), no. 1, 49–66. MR 1025673, DOI 10.1007/BF01195292
Raúl E. Curto, Quadratically hyponormal weighted shifts, Integral Equations Operator Theory 13 (1990), no. 1, 49–66. MR 1025673, DOI 10.1007/BF01195292
Raúl E. Curto and Lawrence A. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, Integral Equations Operator Theory 17 (1993), no. 2, 202–246. MR 1233668, DOI 10.1007/BF01200218
Raúl E. Curto and Lawrence A. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, Integral Equations Operator Theory 17 (1993), no. 2, 202–246. MR 1233668, DOI 10.1007/BF01200218
Raúl E. Curto and Lawrence A. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, Integral Equations Operator Theory 17 (1993), no. 2, 202–246. MR 1233668, DOI 10.1007/BF01200218
Raúl E. Curto, Sang Hoon Lee, and Jasang Yoon, $k$-hyponormality of multivariable weighted shifts, J. Funct. Anal. 229 (2005), no. 2, 462–480. MR 2183156, DOI 10.1016/j.jfa.2005.03.022
Raúl E. Curto and Woo Young Lee, Joint hyponormality of Toeplitz pairs, Mem. Amer. Math. Soc. 150 (2001), no. 712, x+65. MR 1810770, DOI 10.1090/memo/0712
Mary R. Embry, A generalization of the Halmos-Bram criterion for subnormality, Acta Sci. Math. (Szeged) 35 (1973), 61–64. MR 328652
Mary Embry-Wardrop and Alan Lambert, Subnormality for the adjoint of a composition operator on $L^2$, J. Operator Theory 25 (1991), no. 2, 309–318. MR 1203036
G. Exner, On $n$-contractive and $n$-hypercontractive operators, Integral Equations Operator Theory, 56 (2006), 451–468.
G. Exner, I. Jung, and S. Park, On $n$-hypercontractive operators, II, submitted.
Takayuki Furuta, Invitation to linear operators, Taylor & Francis Group, London, 2001. From matrices to bounded linear operators on a Hilbert space. MR 1978629, DOI 10.1201/b16820
Masatoshi Ito and Takeaki Yamazaki, Relations between two inequalities $(B^{\frac r2}A^pB^{\frac r2})^{\frac r{p+r}}\geq B^r$ and $A^p\geq (A^{\frac p2}B^rA^{\frac p2})^{\frac p{p+r}}$ and their applications, Integral Equations Operator Theory 44 (2002), no. 4, 442–450. MR 1942034, DOI 10.1007/BF01193670
Il Bong Jung, Eungil Ko, Chunji Li, and Sang Soo Park, Embry truncated complex moment problem, Linear Algebra Appl. 375 (2003), 95–114. MR 2013458, DOI 10.1016/S0024-3795(03)00617-7
I. Jung, C. Li and S. Park, Complex moment matrices via Halmos-Bram and Embry conditions, J. Korean Math. Soc., to appear.
Il Bong Jung and Chunji Li, A formula for $k$-hyponormality of backstep extensions of subnormal weighted shifts, Proc. Amer. Math. Soc. 129 (2001), no. 8, 2343–2351. MR 1823917, DOI 10.1090/S0002-9939-00-05844-5
Alan Lambert, Hyponormal composition operators, Bull. London Math. Soc. 18 (1986), no. 4, 395–400. MR 838810, DOI 10.1112/blms/18.4.395
Scott McCullough and Vern Paulsen, A note on joint hyponormality, Proc. Amer. Math. Soc. 107 (1989), no. 1, 187–195. MR 972236, DOI 10.1090/S0002-9939-1989-0972236-8
Scott McCullough and Vern Paulsen, $k$-hyponormality of weighted shifts, Proc. Amer. Math. Soc. 116 (1992), no. 1, 165–169. MR 1102858, DOI 10.1090/S0002-9939-1992-1102858-5
M. M. Rao, Conditional measures and applications, Monographs and Textbooks in Pure and Applied Mathematics, vol. 177, Marcel Dekker, Inc., New York, 1993. MR 1234936
Jung Woi Park and Sang Soo Park, On $k$-hyponormal weighted translation semigroups, Bull. Korean Math. Soc. 39 (2002), no. 4, 527–534. MR 1938992, DOI 10.4134/BKMS.2002.39.4.527
J. A. Shohat and J. D. Tamarkin, The Problem of Moments, American Mathematical Society Mathematical Surveys, Vol. I, American Mathematical Society, New York, 1943. MR 0008438, DOI 10.1090/surv/001
R. K. Singh and J. S. Manhas, Composition operators on function spaces, North-Holland Mathematics Studies, vol. 179, North-Holland Publishing Co., Amsterdam, 1993. MR 1246562, DOI 10.1016/S0304-0208(08)71589-5