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CLASSICAL PROPERTIES OF COMPOSITION OPERATORS ON HARDY–ORLICZ SPACES ON PLANAR DOMAINS

Part of: Linear function spaces and their duals

Published online by Cambridge University Press:  29 October 2018

MICHAŁ RZECZKOWSKI
MICHAŁ RZECZKOWSKI*
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, 61-614 Poznan, Poland email rzeczkow@amu.edu.pl
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Abstract

In this paper we study composition operators on Hardy–Orlicz spaces on multiply connected domains whose boundaries consist of finitely many disjoint analytic Jordan curves. We obtain a characterization of order-bounded composition operators. We also investigate weak compactness and the Dunford–Pettis property of these operators.

Keywords

Hardy–Orlicz spacescomposition operatorsHardy spaces on planar domains

MSC classification

Primary: 46E15: Banach spaces of continuous, differentiable or analytic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 107 , Issue 2 , October 2019 , pp. 256 - 271
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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Footnotes

This research was supported by National Science Centre research grant 2015/19/N/ST1/00845.

References

Cowen, C. and MacCluer, B., Composition Operators on Spaces of Analytic Functions (CRC Press, Boca Raton, FL, 1995).Google Scholar
Fisher, S. D., Function Theory on Planar Domains (Wiley, New York, 1983).Google Scholar
Fisher, S. D. and Shapiro, J. E., ‘The essential norm of composition operators on a planar domain’, Illinois J. Math. 43 (1999), 113–130.Google Scholar
Forelli, F., ‘The isometries of H p ’, Canad. J. Math. 16 (1964), 721–728.Google Scholar
Garnett, J. B. and Marshall, D. E., Harmonic Measure (Cambridge University Press, New York, 2005).Google Scholar
Krasnosielskii, M. A. and Rutycki, J. B., Convex Functions and Orlicz Spaces (Noordhoff, Groningen, 1961).Google Scholar
Lefèvre, P., Li, D., Queffélec, H. and Rodríguez-Piazza, L., ‘A criterion of weak compactness for operators on subspaces of Orlicz spaces’, J. Funct. Spaces Appl. 6(3) (2008), 277–292.Google Scholar
Lefèvre, P., Li, D., Queffélec, H. and Rodríguez-Piazza, L., ‘Some examples of compact composition operators on H 2 ’, J. Funct. Anal. 255(11) (2008), 3098–3124.Google Scholar
Lefèvre, P., Li, D., Queffélec, H. and Rodríguez-Piazza, L., ‘Compact composition operators on H 2 and Hardy–Orlicz spaces’, J. Math. Anal. Appl. 354(1) (2009), 360–371.Google Scholar
Lefèvre, P., Li, D., Queffélec, H. and Rodríguez-Piazza, L., ‘Composition operators on Hardy–Orlicz spaces’, Mem. Amer. Math. Soc. 207(974) (2010), 1–74.Google Scholar
Lefèvre, P., Li, D., Quefféleci, H. and Rodríguez-Piazza, L., ‘Nevanlinna counting function and Carleson function of analytic maps’, Math. Ann. 351 (2011), 305–326.Google Scholar
MacCluer, B. D., ‘Compact composition operators on H p (B N)’, Michigan Math. J. 32 (1985), 237–248.Google Scholar
Mastyło, M. and Mleczko, P., ‘Solid hulls of quasi-Banach spaces of analytic functions and interpolation’, Nonlinear Anal. 73(1) (2010), 84–98.Google Scholar
Rao, M. M. and Ren, Z. D., Theory of Orlicz Spaces, Pure and Applied Mathematics, 146 (Marcel Dekker, New York, 1991).Google Scholar
Rudin, W., ‘Analytic functions of class H p ’, Trans. Amer. Math. Soc. 78 (1955), 46–66.Google Scholar
Rzeczkowski, M., ‘Composition operators on Hardy–Orlicz spaces on planar domains’, Ann. Acad. Sci. Fenn. Math. 42 (2017), 593–609.Google Scholar
Sarason, D., ‘The H p spaces of an annulus’, Mem. Amer. Math. Soc. 56 (1965), 1–78.Google Scholar
Shapiro, J. H., ‘The essential norm of a composition operator’, Ann. of Math. (2) 125 (1987), 375–404.Google Scholar
Shapiro, J. H., Composition Operators and Classical Function Theory (Springer, New York, 1991).Google Scholar