Compact composition operators on 𝐿¹

Shapiro, Joel H.; Sundberg, Carl

doi:10.1090/S0002-9939-1990-0994787-0

Compact composition operators on $L\sp 1$

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
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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 ${H^2}$ .

[1] 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
[2] Peter L. Duren, Theory of 𝐻^{𝑝} spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
[3] M. Essén, D. F. Shea, and C. S. Stanton, A value-distribution criterion for the class 𝐿𝑙𝑜𝑔𝐿, and some related questions, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 4, 127–150 (English, with French summary). MR 812321
[4] 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
[5] Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR 0133008
[6] J. E. Littlewood, On inequalities in the theory of functions, Proc. London Math. Soc. (2) 23 (1925), 481-519.
[7] D. J. Newman, The nonexistence of projections from 𝐿¹ to 𝐻¹, Proc. Amer. Math. Soc. 12 (1961), 98–99. MR 120524, https://doi.org/10.1090/S0002-9939-1961-0120524-X
[8] 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
[9] 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
[10] J. H. Shapiro and P. D. Taylor, Compact, nuclear, and Hilbert-Schmidt composition operators on 𝐻², Indiana Univ. Math. J. 23 (1973/74), 471–496. MR 326472, https://doi.org/10.1512/iumj.1973.23.23041
[11] Joel H. Shapiro, The essential norm of a composition operator, Ann. of Math. (2) 125 (1987), no. 2, 375–404. MR 881273, https://doi.org/10.2307/1971314
[12] Joel H. Shapiro and Carl Sundberg, Isolation amongst the composition operators, Pacific J. Math. 145 (1990), no. 1, 117–152. MR 1066401
[13] C. S. Stanton, Riesz mass and growth problems for subharmonic functions, Thesis, University of Wisconsin, Madison, 1982.
[14] Charles S. Stanton, Counting functions and majorization for Jensen measures, Pacific J. Math. 125 (1986), no. 2, 459–468. MR 863538