Proceedings of the American Mathematical Society

Author(s): Rajeev Kumar; Romesh Kumar
Journal: Proc. Amer. Math. Soc. 133 (2005), 2109-2118.
MSC (2000): Primary 47B33, 46E30; Secondary 47B07, 46B70
Posted: February 15, 2005
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Abstract: We study the boundedness and the compactness of composition operators on some Banach function spaces such as absolutely continuous Banach function spaces on a $\sigma$ -finite measure space, Lorentz function spaces on a $\sigma$ -finite measure space and rearrangement invariant spaces on a resonant measure space. In addition, we study some properties of the spectra of a composition operator on the general Banach function spaces.

1.: C. Bennett and R. Sharpley, Interpolation of operators, Pure and Applied Mathematics 129, Academic Press, London 1988. MR 0928802 (89e:46001)
2.: P. S. Bourdon and J. H. Shapiro, Cyclic phenomenon for composition operators, Memoirs of the AMS, 1996.
3.: P. L. Butzer and H. Berens, Semigroups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen Band 145, Springer-Verlag, New York 1967. MR 0230022 (37:5588)
4.: J. T. Campbell and J. E. Jamison, The analysis of composition operators on and the Hopf decomposition, J. Math. Anal. Appl. 159, (1991), 520-531. MR 1120950 (92h:47036)
5.: C. C. Cowen and B. D. MacCluer, Composition operators on the spaces of analytic functions, CRC Press, Boca Raton, 1995. MR 1397026 (97i:47056)
6.: Y. Cui, H. Hudzik, R. Kumar and L. Maligranda, Composition operators in Orlicz spaces, J. Austral. Math. Soc. 76 (2004), no. 2, 189-206. MR 2041244 (2004m:47046)
7.: U. Krengel, Ergodic theorems, Walter de Gruyter, Berlin, 1985. MR 0797411 (87i:28001)
8.: S. G. Krein, Ju. I. Petunin and E. M. Semenov, Interpolation of linear operators, AMS Translation of Math. Monographs 54, Providence, RI 1982.
9.: Rajeev Kumar and Romesh Kumar, Composition operators in Lorentz spaces, (Preprint)
10.: R. Kumar, Composition operators on Orlicz spaces, Integral Equations and Operator Theory 29 (1997), 17-22. MR 1466857 (98k:47056)
11.: A. Lambert, Hyponormal composition operators, London Math. Soc. 18, (1986), 395-400. MR 0838810 (87h:47059)
12.: J. Lindenstrauss and L. Tzafriri, Classical Banach spaces II, Function Spaces, Springer Verlag, Berlin-New York 1979. MR 0540367 (81c:46001)
13.: R. Nagel and K. J. Engel, One parameter semigroups for linear evolution equations, Graduate Texts in Math., Springer-Verlag, New York 2000. MR 1721989 (2000i:47075)
14.: S. Petrovic, A note on composition operators, Mat. Vestnik 40 (1988), 147-151. MR 0979931 (90c:47056)
15.: M. M. Rao, Applications of Orlicz spaces, Marcel Dekker, New York 2002.
16.: W. C. Ridge, Characterisation of abstract composition operators, Proc. Amer. Math. Soc. 45 (1974), 393-396. MR 0346585 (49:11310)
17.: W. C. Ridge, Spectrum of composition operators, Proc. Amer. Math. Soc. 37 (1973), 121-127. MR 0306457 (46:5583)
18.: W. Rudin, Real and complex analysis, 3rd edition, New York 1987.
19.: J. H. Shapiro, Composition operators and classical function theory, Springer-Verlag, New York 1993. MR 1237406 (94k:47049)
20.: R. K. Singh and A. Kumar, Compact composition operators, J. Austral. Math. Soc. 28 (1979), 309-314. MR 0557280 (81g:47033)
21.: R. K. Singh and J. S. Manhas, Composition operators on function spaces, North Holland Math. Studies 179, Amsterdam 1993. MR 1246562 (95d:47036)
22.: H. Takagi and K. Yokouchi, Multiplication and composition operators between two -spaces, in: Function Spaces (Edwardsville, IL 1998), Contemp. Math. 232, Amer. Math. Soc., Providence, RI 1999, 321-338. MR 1678344 (2000g:47037)
23.: X. M. Xu, Compact composition operators on $L^p(X,\Sigma,\mu)$ , Adv. in Math. (China) 20 (1991), 221-225 (Chinese). MR 1121561 (92e:47057)

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47B33, 46E30, 47B07, 46B70

Rajeev Kumar
Affiliation: Department of Mathematics, University of Jammu, Jammu-180 006, India
Email: raj1k2@yahoo.co.in

Romesh Kumar
Affiliation: Department of Mathematics, University of Jammu, Jammu-180 006, India
Email: romesh_jammu@yahoo.com

DOI: 10.1090/S0002-9939-05-07798-1
PII: S 0002-9939(05)07798-1
Keywords: Banach function spaces, compact operators, composition operators, Lorentz spaces, measurable transformation, rearrangement invariant spaces
Received by editor(s): January 24, 2004
Received by editor(s) in revised form: April 5, 2004
Posted: February 15, 2005
Additional Notes: The first author was supported in part by CSIR Grant \#9(96)100/2002-EMR-I, dated--13-5-2002).
Communicated by: Joseph A. Ball
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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