Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On compactness of composition operators in Hardy spaces of several variables


Authors: Song-Ying Li and Bernard Russo
Journal: Proc. Amer. Math. Soc. 123 (1995), 161-171
MSC: Primary 47B38; Secondary 32A35, 47B07
DOI: https://doi.org/10.1090/S0002-9939-1995-1213865-9
MathSciNet review: 1213865
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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


References [Enhancements On Off] (What's this?)

  • [1] J. A. Cima and W. Wogen, Unbounded composition operators on $ {H^2}({B_n})$, Proc. Amer. Math. Soc. 99 (1987), 477-483. MR 875384 (88d:32009)
  • [2] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-643. MR 0447954 (56:6264)
  • [3] M. Christ and D. Geller, Singular integral characterizations of Hardy spaces on homogeneous groups, Duke Math. J. 51 (1984), 547-598. MR 757952 (86g:43007a)
  • [4] J. B. Conway, A course in functional analysis, 2nd ed., Springer-Verlag, New York, 1990. MR 1070713 (91e:46001)
  • [5] C. C. Cowen, Composition operators on Hilbert spaces of analytic functions: A status report, Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 131-145. MR 1077383 (91m:47043)
  • [6] N. Dunford and J. T. Schwartz, Linear operator, Part I, Interscience, New York, 1958.
  • [7] L. Hörmander, $ {L^p}$-estimates for pluri-harmonic functions, Math. Scand. 20 (1967), 65-78. MR 0234002 (38:2323)
  • [8] C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1-65. MR 0350069 (50:2562)
  • [9] C. Fefferman and E. M. Stein, $ {H^p}$ spaces of several variables, Acta Math. 129 (1972), 137-193. MR 0447953 (56:6263)
  • [10] F. Jafari, Carleson measures in Hardy and weighted Bergman spaces of poly discs, Proc. Amer. Math. Soc. 112 (1991), 771-781. MR 1039533 (91j:47032)
  • [11] N. Kerzman, The Bergman kernel function. Differentiability at the boundary, Math. Ann. 195 (1972), 149-158. MR 0294694 (45:3762)
  • [12] S. G. Krantz, Function theory of several complex variables, 2nd ed., Wadsworth, Belmont, CA, 1992. MR 1162310 (93c:32001)
  • [13] S. G. Krantz and S.-Y. Li, A note on Hardy spaces and functions of bounded mean oscillation on domains in $ {{\mathbf{C}}^n}$, Michigan Math. J. (to appear). MR 1260608 (95f:32008)
  • [14] -, On decomposition theorems for Hardy spaces on domains in $ {{\mathbf{C}}^n}$ and applications, preprint.
  • [15] B. MacCluer, Spectra of composition operators on $ {H^p}({B_N})$, Analysis 4 (1984), 87-103. MR 775548 (86e:47038)
  • [16] -, Compact composition operators on $ {H^p}({B_N})$, Michigan Math. J. 32 (1985), 237-248.
  • [17] A. Nagel, E. M. Stein, and S. Wainger, Boundary behavior of functions holomorphic in domains of finite type, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), 6596-6599. MR 634936 (82k:32027)
  • [18] A. Nagel, J. P. Rosay, E. M. Stein, and S. Wainger, Estimates for the Bergman and Szegö kernels in $ {{\mathbf{C}}^2}$, Ann. of Math. (2) 129 (1989), 113-149. MR 979602 (90g:32028)
  • [19] J. H. Shapiro, The essential norm of a composition operator, Ann. of Math. (2) 125 (1987), 375-404. MR 881273 (88c:47058)
  • [20] D. Sarason, Weak compactness of holomorphic composition operators on $ {H^1}$, preprint.
  • [21] J. H. Shapiro and C. Sundberg, Compact composition operators on $ {L^1}$, Proc. Amer. Math. Soc. 108 (1992), 443-449. MR 994787 (90d:47035)
  • [22] J. H. Shapiro and P. D. Taylor, Compact nuclear, and Hilbert Schmidt composition operators on $ {H^2}$, Indiana Univ. Math. J. 23 (1973), 471-496. MR 0326472 (48:4816)
  • [23] E. M. Stein, Boundary behavior of holomorphic functions of several complex variables, Princeton Univ. Press, Princeton, NJ, 1972. MR 0473215 (57:12890)
  • [24] W. Wogen, Composition operators acting on spaces of holomorphic functions on domains in $ {{\mathbf{C}}^n}$, Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 361-366. MR 1077457 (91k:47069)
  • [25] R. R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their application to harmonic analysis, J. Funct. Anal. 62 (1985), 304-335. MR 791851 (86i:46029)
  • [26] B. MacCluer and J. H. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38 (1986), 878-906. MR 854144 (87h:47048)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47B38, 32A35, 47B07

Retrieve articles in all journals with MSC: 47B38, 32A35, 47B07


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1213865-9
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society