Compact composition operators on $L^ 1$
Authors:
Joel H. Shapiro and Carl Sundberg
Journal:
Proc. Amer. Math. Soc. 108 (1990), 443-449
MSC:
Primary 47B38; Secondary 30D55, 47B05
DOI:
https://doi.org/10.1090/S0002-9939-1990-0994787-0
MathSciNet review:
994787
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: The composition operator induced by a holomorphic self-map of the unit disc is compact on ${L^1}$ of the unit circle if and only if it is compact on the Hardy space ${H^2}$ of the disc. This answers a question posed by Donald Sarason: it proves that Sarason’s integral condition characterizing compactness on ${L^1}$ is equivalent to the asymptotic condition on the Nevanlinna counting function which characterizes compactness on ${H^2}$.
- Carl C. Cowen, Composition operators on Hilbert spaces of analytic functions: a status report, Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 131–145. MR 1077383, DOI https://doi.org/10.1016/j.jpaa.2009.05.015
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
- M. Essén, D. F. Shea, and C. S. Stanton, A value-distribution criterion for the class $L\,{\rm log}\,L$, and some related questions, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 4, 127–150 (English, with French summary). MR 812321
- W. K. Hayman and P. B. Kennedy, Subharmonic functions. Vol. I, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. London Mathematical Society Monographs, No. 9. MR 0460672
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR 0133008 J. E. Littlewood, On inequalities in the theory of functions, Proc. London Math. Soc. (2) 23 (1925), 481-519.
- D. J. Newman, The nonexistence of projections from $L^{1}$ to $H^{1}$, Proc. Amer. Math. Soc. 12 (1961), 98–99. MR 120524, DOI https://doi.org/10.1090/S0002-9939-1961-0120524-X
- Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. McGraw-Hill Series in Higher Mathematics. MR 0344043
- Donald Sarason, Composition operators as integral operators, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 545–565. MR 1044808
- 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 https://doi.org/10.1512/iumj.1973.23.23041
- Joel H. Shapiro, The essential norm of a composition operator, Ann. of Math. (2) 125 (1987), no. 2, 375–404. MR 881273, DOI https://doi.org/10.2307/1971314
- Joel H. Shapiro and Carl Sundberg, Isolation amongst the composition operators, Pacific J. Math. 145 (1990), no. 1, 117–152. MR 1066401 C. S. Stanton, Riesz mass and growth problems for subharmonic functions, Thesis, University of Wisconsin, Madison, 1982.
- Charles S. Stanton, Counting functions and majorization for Jensen measures, Pacific J. Math. 125 (1986), no. 2, 459–468. MR 863538
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47B38, 30D55, 47B05
Retrieve articles in all journals with MSC: 47B38, 30D55, 47B05
Additional Information
Keywords:
Compact composition operator,
Nevanlinna counting function,
Riesz mass
Article copyright:
© Copyright 1990
American Mathematical Society
