Compact composition operators on the Nevanlinna class
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- by Jun Soo Choa and Hong Oh Kim PDF
- Proc. Amer. Math. Soc. 125 (1997), 145-151 Request permission
Abstract:
In this paper we prove that the composition operator induced by a holomorphic self-map of the unit disc is compact on the Nevanlinna class if and only if it is compact on the Hardy space $H^2$.References
- Carl C. Cowen, Composition operators on $H^{2}$, J. Operator Theory 9 (1983), no. 1, 77–106. MR 695941
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
- John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
- Barbara D. MacCluer, Compact composition operators on $H^p(B_N)$, Michigan Math. J. 32 (1985), no. 2, 237–248. MR 783578, DOI 10.1307/mmj/1029003191
- M. I. Masri, Compact composition operators on the Nevanlinna and Smirnov classes, Thesis, University of North Carolina, Chapel Hill, 1985.
- Barbara D. MacCluer and Joel H. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38 (1986), no. 4, 878–906. MR 854144, DOI 10.4153/CJM-1986-043-4
- Eric A. Nordgren, Composition operators, Canadian J. Math. 20 (1968), 442–449. MR 223914, DOI 10.4153/CJM-1968-040-4
- H. J. Schwartz, Composition operators on $H^p$, Thesis, University of Toledo, 1969.
- Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1237406, DOI 10.1007/978-1-4612-0887-7
- Joel H. Shapiro, The essential norm of a composition operator, Ann. of Math. (2) 125 (1987), no. 2, 375–404. MR 881273, DOI 10.2307/1971314
- J. H. Shapiro and A. L. Shields, Unusual topological properties of the Nevanlinna class, Amer. J. of Math. 97(4), 915–936.
- J. H. Shapiro and P. D. Taylor, Compact, nuclear, and Hilbert-Schmidt composition operators on $H^{2}$, Indiana Univ. Math. J. 23 (1973/74), 471–496. MR 326472, DOI 10.1512/iumj.1973.23.23041
- C. S. Stanton, Riesz mass and growth problems for subharmonic functions, Thesis, Univ. of Wisconsin, Madison, 1982.
Additional Information
- Jun Soo Choa
- Affiliation: Department of Mathematics Education, Sung Kyun Kwan University, Jongro-Gu, Seoul 110-745, Korea
- Email: jschoa@yurim.skku.ac.kr
- Hong Oh Kim
- Affiliation: Department of Mathematics, Korea Advanced Institute of Science and Technology, 373-1 Yusung-gu Gusung-dong, Taejun 305-701, Korea
- Received by editor(s): May 2, 1995
- Received by editor(s) in revised form: June 30, 1995
- Additional Notes: Supported in part by TGRC, and Korean ministry of Education, BSRI-95-1420
- Communicated by: Theodore W. Gamelin
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 145-151
- MSC (1991): Primary 47B05, 47B38; Secondary 30D55
- DOI: https://doi.org/10.1090/S0002-9939-97-03534-X
- MathSciNet review: 1346966