Compact composition operators on

Authors:
Joel H. Shapiro and Carl Sundberg

Journal:
Proc. Amer. Math. Soc. **108** (1990), 443-449

MSC:
Primary 47B38; Secondary 30D55, 47B05

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 of the unit circle if and only if it is compact on the Hardy space of the disc. This answers a question posed by Donald Sarason: it proves that Sarason's integral condition characterizing compactness on is equivalent to the asymptotic condition on the Nevanlinna counting function which characterizes compactness on .

**[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**0120524**, 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**0326472****[11]**Joel H. Shapiro,*The essential norm of a composition operator*, Ann. of Math. (2)**125**(1987), no. 2, 375–404. MR**881273**, 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**

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

DOI:
https://doi.org/10.1090/S0002-9939-1990-0994787-0

Keywords:
Compact composition operator,
Nevanlinna counting function,
Riesz mass

Article copyright:
© Copyright 1990
American Mathematical Society