Compact composition operators on the Smirnov class

Choa, Jun; Kim, Hong; Shapiro, Joel

doi:10.1090/S0002-9939-99-05239-9

Compact composition operators on the Smirnov class
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by Jun Soo Choa, Hong Oh Kim and Joel H. Shapiro PDF

Proc. Amer. Math. Soc. 128 (2000), 2297-2308 Request permission

Abstract:

We show that a composition operator on the Smirnov class $N^+$ is compact if and only if it is compact on some (equivalently: every) Hardy space $H^p$ for $0<p<\infty$. Along the way we show that for composition operators on $N^+$ both the formally weaker notion of boundedness, and a formally stronger notion we call metric compactness, are equivalent to compactness.

References

Jun Soo Choa and Hong Oh Kim, Compact composition operators on the Nevanlinna class, Proc. Amer. Math. Soc. 125 (1997), no. 1, 145–151. MR 1346966, DOI 10.1090/S0002-9939-97-03534-X
Joseph A. Cima and Alec L. Matheson, Essential norms of composition operators and Aleksandrov measures, Pacific J. Math. 179 (1997), no. 1, 59–64. MR 1452525, DOI 10.2140/pjm.1997.179.59
Carl C. Cowen and Barbara D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397026
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
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
Farhad Jafari, Barbara D. MacCluer, Carl C. Cowen, and A. Duane Porter (eds.), Studies on composition operators, Contemporary Mathematics, vol. 213, American Mathematical Society, Providence, RI, 1998. MR 1601044, DOI 10.1090/conm/213
J. L. Kelley and Isaac Namioka, Linear topological spaces, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. MR 0166578, DOI 10.1007/978-3-662-41914-4
M. I. Masri, Compact Composition Operators on the Nevanlinna and Smirnov Classes, Thesis, Univ. of North Carolina at Chapel Hill 1985.
Niro Yanagihara and Yoshikuni Nakamura, Composition operators on the class $N^{+}$, TRU Math. 14 (1978), no. 2, 9–16. MR 526517
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
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
H. J. Schwartz, Composition operators on $H^p$, Thesis, Univ. of Toledo 1969.
Jonathan E. Shapiro, Aleksandrov measures used in essential norm inequalities for composition operators, J. Operator Th., to appear.
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
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 and Allen L. Shields, Unusual topological properties of the Nevanlinna class, Amer. J. Math. 97 (1975), no. 4, 915–936. MR 390227, DOI 10.2307/2373681
Joel H. Shapiro and Carl Sundberg, Compact composition operators on $L^1$, Proc. Amer. Math. Soc. 108 (1990), no. 2, 443–449. MR 994787, DOI 10.1090/S0002-9939-1990-0994787-0
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
V. I. Smirnov, Sur les valeurs limits des fonctions, regulière à l’intérieur d’un cercle, J. Soc. Phys. Math. Léningrade 2, No. 2 (1929), 22–37.
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Niro Yanagihara, Multipliers and linear functionals for the class $N^{+}$, Trans. Amer. Math. Soc. 180 (1973), 449–461. MR 338382, DOI 10.1090/S0002-9947-1973-0338382-X