On compactness of composition operators in Hardy spaces of several variables

On compactness of composition operators in Hardy spaces of several variables
HTML articles powered by AMS MathViewer

Proc. Amer. Math. Soc. 123 (1995), 161-171

DOI: https://doi.org/10.1090/S0002-9939-1995-1213865-9

PDF | Request permission

Characterizations of compactness are given for holomorphic composition operators on Hardy spaces of a strongly pseudoconvex domain.

References

J. A. Cima and W. R. Wogen, Unbounded composition operators on $H^2(B_2)$, Proc. Amer. Math. Soc. 99 (1987), no. 3, 477–483. MR 875384, DOI 10.1090/S0002-9939-1987-0875384-4
Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI 10.1090/S0002-9904-1977-14325-5
Michael Christ and Daryl Geller, Singular integral characterizations of Hardy spaces on homogeneous groups, Duke Math. J. 51 (1984), no. 3, 547–598. MR 757952, DOI 10.1215/S0012-7094-84-05127-5
John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713
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 10.1090/pspum/051.1/1077383

Linear operator

Lars Hörmander, $L^{p}$ estimates for (pluri-) subharmonic functions, Math. Scand. 20 (1967), 65–78. MR 234002, DOI 10.7146/math.scand.a-10821
Charles Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1–65. MR 350069, DOI 10.1007/BF01406845
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
F. Jafari, Carleson measures in Hardy and weighted Bergman spaces of polydiscs, Proc. Amer. Math. Soc. 112 (1991), no. 3, 771–781. MR 1039533, DOI 10.1090/S0002-9939-1991-1039533-0
Norberto Kerzman, The Bergman kernel function. Differentiability at the boundary, Math. Ann. 195 (1972), 149–158. MR 294694, DOI 10.1007/BF01419622
Steven G. Krantz, Function theory of several complex variables, 2nd ed., The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. MR 1162310
Steven G. Krantz and Song-Ying Li, A note on Hardy spaces and functions of bounded mean oscillation on domains in $\textbf {C}^n$, Michigan Math. J. 41 (1994), no. 1, 51–71. MR 1260608, DOI 10.1307/mmj/1029004914

On decomposition theorems for Hardy spaces on domains in

and applications

Barbara D. MacCluer, Spectra of compact composition operators on $H^p(B_N)$, Analysis 4 (1984), no. 1-2, 87–103. MR 775548, DOI 10.1524/anly.1984.4.12.87

Compact composition operators on

32

Alexander Nagel, Elias M. Stein, and Stephen Wainger, Boundary behavior of functions holomorphic in domains of finite type, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), no. 11, 6596–6599. MR 634936, DOI 10.1073/pnas.78.11.6596
A. Nagel, J.-P. Rosay, E. M. Stein, and S. Wainger, Estimates for the Bergman and Szegő kernels in $\textbf {C}^2$, Ann. of Math. (2) 129 (1989), no. 1, 113–149. MR 979602, DOI 10.2307/1971487
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

Weak compactness of holomorphic composition operators on

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
E. M. Stein, Boundary behavior of holomorphic functions of several complex variables, Mathematical Notes, No. 11, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. MR 0473215
Warren R. Wogen, Composition operators acting on spaces of holomorphic functions on domains in $\textbf {C}^n$, Operator theory: operator algebras and applications, Part 2 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 361–366. MR 1077457, DOI 10.1090/pspum/051.2/1077457
R. R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), no. 2, 304–335. MR 791851, DOI 10.1016/0022-1236(85)90007-2
Barbara D. MacCluer and Joel H. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38 (1986), no. 4, 878–906. MR 854144, DOI 10.4153/CJM-1986-043-4