Completely continuous composition operators

Cima, Joseph A.; Matheson, Alec

doi:10.1090/S0002-9947-1994-1257642-5

Completely continuous composition operators
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by Joseph A. Cima and Alec Matheson PDF

Trans. Amer. Math. Soc. 344 (1994), 849-856 Request permission

Abstract:

A composition operator ${T_b}f = f \circ b$ is completely continuous on ${H^1}$ if and only if $|b| < 1$ a.e. If the adjoint operator $T_b^\ast$ is completely continuous on VMOA, then ${T_b}$ is completely continuous on ${H^1}$. Examples are given to show that the converse fails in general. Two results are given concerning the relationship between the complete continuity of an operator and of its adjoint in the presence of certain separability conditions on the underlying Banach space.

References

Opérations linéaires

J. Bourgain, Dunford-Pettis operators on $L^{1}$ and the Radon-Nikodým property, Israel J. Math. 37 (1980), no. 1-2, 34–47. MR 599300, DOI 10.1007/BF02762866
J. Bourgain, Dunford-Pettis operators on $L^{1}$ and the Radon-Nikodým property, Israel J. Math. 37 (1980), no. 1-2, 34–47. MR 599300, DOI 10.1007/BF02762866
John B. Conway, A course in functional analysis, Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1985. MR 768926, DOI 10.1007/978-1-4757-3828-5
Joe Diestel, A survey of results related to the Dunford-Pettis property, Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, N.C., 1979) Contemp. Math., vol. 2, Amer. Math. Soc., Providence, R.I., 1980, pp. 15–60. MR 621850
Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
Leonard E. Dor, On sequences spanning a complex $l_{1}$ space, Proc. Amer. Math. Soc. 47 (1975), 515–516. MR 358308, DOI 10.1090/S0002-9939-1975-0358308-X
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
Haskell P. Rosenthal, A characterization of Banach spaces containing $l^{1}$, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411–2413. MR 358307, DOI 10.1073/pnas.71.6.2411
Haskell P. Rosenthal, Point-wise compact subsets of the first Baire class, Amer. J. Math. 99 (1977), no. 2, 362–378. MR 438113, DOI 10.2307/2373824
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

Weak compactness of holomorphic composition operators

Joel H. Shapiro, Compact composition operators on spaces of boundary-regular holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), no. 1, 49–57. MR 883400, DOI 10.1090/S0002-9939-1987-0883400-9
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 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