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

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]**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*, J. Operator Theory,**25**(1991), 309-318. MR**1203036 (94f:47028)****[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]**T. Furuta,*Invitation to linear operators*, Taylor & Francis Inc., 2001. MR**1978629 (2004b:47001)****[16]**M. Ito and T. Yamazaki,*Relations between two inequalities*and*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**-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. 1**07**(1989), 187-195. MR**972236 (90a:47062)****[22]**,*-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**-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)**

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