## On compactness of composition operators in Hardy spaces of several variables

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- by Song-Ying Li and Bernard Russo PDF
- Proc. Amer. Math. Soc.
**123**(1995), 161-171 Request permission

## Abstract:

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 10.1016/j.jpaa.2009.05.015
N. Dunford and J. T. Schwartz, - 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
—, - 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
—, - 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
D. Sarason, - 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

*Linear operator*, Part I, Interscience, New York, 1958.

*On decomposition theorems for Hardy spaces on domains in*${{\mathbf {C}}^n}$

*and applications*, preprint.

*Compact composition operators on*${H^p}({B_N})$, Michigan Math. J.

**32**(1985), 237-248.

*Weak compactness of holomorphic composition operators on*${H^1}$, preprint.

## Additional Information

- © Copyright 1995 American Mathematical Society
- 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