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

Published electronically:
June 19, 2007

MathSciNet review:
2341946

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Several classes have been considered to study the weak subnormalities of Hilbert space operators. One of them is -hypnormality, which comes from the Bram-Halmos criterion for subnormal operators. In this note we consider -hyponormality, which is the parallel version corresponding to the Embry characterization for subnormal operators. We characterize -hyponormality of composition operators via -th Radon-Nikodym derivatives and present some examples to distinguish the classes.

**[1]**Jim Agler,*Hypercontractions and subnormality*, J. Operator Theory**13**(1985), no. 2, 203–217. MR**775993****[2]**C. Burnap and I. Jung,*Composition operators with weak hyponormality*, J. Math. Anal. Appl., to appear.**[3]**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****[4]**Raúl E. Curto,*Quadratically hyponormal weighted shifts*, Integral Equations Operator Theory**13**(1990), no. 1, 49–66. MR**1025673**, 10.1007/BF01195292**[5]**Raúl E. Curto,*Quadratically hyponormal weighted shifts*, Integral Equations Operator Theory**13**(1990), no. 1, 49–66. MR**1025673**, 10.1007/BF01195292**[6]**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**, 10.1007/BF01200218**[7]**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**, 10.1007/BF01200218**[8]**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**, 10.1007/BF01200218**[9]**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**, 10.1016/j.jfa.2005.03.022**[10]**Raúl E. Curto and Woo Young Lee,*Joint hyponormality of Toeplitz pairs*, Mem. Amer. Math. Soc.**150**(2001), no. 712, x+65. MR**1810770**, 10.1090/memo/0712**[11]**Mary R. Embry,*A generalization of the Halmos-Bram criterion for subnormality*, Acta Sci. Math. (Szeged)**35**(1973), 61–64. MR**0328652****[12]**Mary Embry-Wardrop and Alan Lambert,*Subnormality for the adjoint of a composition operator on 𝐿²*, J. Operator Theory**25**(1991), no. 2, 309–318. MR**1203036****[13]**G. Exner,*On*-*contractive and*-*hypercontractive operators*, Integral Equations Operator Theory,**56**(2006), 451-468.**[14]**G. Exner, I. Jung, and S. Park,*On**-hypercontractive operators*, II, submitted.**[15]**Takayuki Furuta,*Invitation to linear operators*, Taylor & Francis, Ltd., London, 2001. From matrices to bounded linear operators on a Hilbert space. MR**1978629****[16]**Masatoshi Ito and Takeaki Yamazaki,*Relations between two inequalities (𝐵^{\frac𝑟2}𝐴^{𝑝}𝐵^{\frac𝑟2})^{\frac𝑟𝑝+𝑟}≥𝐵^{𝑟} and 𝐴^{𝑝}≥(𝐴^{\frac𝑝2}𝐵^{𝑟}𝐴^{\frac𝑝2})^{\frac𝑝𝑝+𝑟} and their applications*, Integral Equations Operator Theory**44**(2002), no. 4, 442–450. MR**1942034**, 10.1007/BF01193670**[17]**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**, 10.1016/S0024-3795(03)00617-7**[18]**I. Jung, C. Li and S. Park,*Complex moment matrices via Halmos-Bram and Embry conditions,*J. Korean Math. Soc., to appear.**[19]**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**, 10.1090/S0002-9939-00-05844-5**[20]**Alan Lambert,*Hyponormal composition operators*, Bull. London Math. Soc.**18**(1986), no. 4, 395–400. MR**838810**, 10.1112/blms/18.4.395**[21]**Scott McCullough and Vern Paulsen,*A note on joint hyponormality*, Proc. Amer. Math. Soc.**107**(1989), no. 1, 187–195. MR**972236**, 10.1090/S0002-9939-1989-0972236-8**[22]**Scott McCullough and Vern Paulsen,*𝑘-hyponormality of weighted shifts*, Proc. Amer. Math. Soc.**116**(1992), no. 1, 165–169. MR**1102858**, 10.1090/S0002-9939-1992-1102858-5**[23]**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****[24]**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**, 10.4134/BKMS.2002.39.4.527**[25]**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****[26]**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**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
47B20,
47B33,
47A63

Retrieve articles in all journals with MSC (2000): 47B20, 47B33, 47A63

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