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HOLOMORPHIC SUPERPOSITION OPERATORS BETWEEN BANACH FUNCTION SPACES

Part of: Nonlinear operators and their properties Measures, integration, derivative, holomorphy Linear function spaces and their duals

Published online by Cambridge University Press:  01 April 2014

CHRISTOPHER BOYD  and
PILAR RUEDA
CHRISTOPHER BOYD*
Affiliation:
School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland
PILAR RUEDA
Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Valencia, 46100 Burjasot, Valencia, Spain email [email protected]
*
✉[email protected]
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Abstract

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We prove that for a large class of Banach function spaces continuity and holomorphy of superposition operators are equivalent and that bounded superposition operators are continuous. We also use techniques from infinite dimensional holomorphy to establish the boundedness of certain superposition operators. Finally, we apply our results to the study of superposition operators on weighted spaces of holomorphic functions and the $F(p, \alpha , \beta )$ spaces of Zhao. Some independent properties on these spaces are also obtained.

Keywords

superposition operatorsweighted spaces of holomorphic functionscontinuity

MSC classification

Secondary: 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) 46G20: Infinite-dimensional holomorphy 46E15: Banach spaces of continuous, differentiable or analytic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 96 , Issue 2 , April 2014 , pp. 186 - 197
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Aulaskari, R., Xiao, J. and Zhao, R., ‘On subspaces and subsets of BMOA and UBC’, Analysis 15 (1995), 101–121.CrossRefGoogle Scholar
Álvarez, V., Márquez, M. A. and Vukotić, D., ‘Superposition operators between the Bloch space and Bergman spaces’, Ark. Mat. 42 (2) (2004), 205–216.Google Scholar
Aron, R. M. and Berner, P., ‘A Hahn–Banach extension theorem for analytic mappings’, Bull. Math. Soc. France 106 (1978), 3–24.Google Scholar
Aron, R., Galindo, P. and Lindström, M., ‘Compact homomorphisms between algebras of analytic functions’, Studia Math. 123 (3) (1997), 235–247.Google Scholar
Aron, R., Galindo, P. and Lindström, M., ‘Connected components in the space of composition operators in ${H}^{\infty } $ functions of many variables’, Integral Equations Operator Theory 45 (1) (2003), 1–14.Google Scholar
Aron, R. and Schottenloher, M., ‘Compact holomorphic mappings on Banach spaces and the approximation property’, J. Funct. Anal. 21 (1) (1976), 7–30.CrossRefGoogle Scholar
Bierstedt, K. D., Bonet, J. and Taskinen, J., ‘Associated weights and spaces of holomorphic functions’, Studia Math. 127 (2) (1998), 137–168.Google Scholar
Bierstedt, K. D. and Summers, W. H., ‘Biduals of weighted Banach spaces of analytic functions’, J. Aust. Math. Soc. Ser. A 54 (1993), 70–79.CrossRefGoogle Scholar
Boas, R. P. Jr, Entire Functions (Academic Press Inc, New York, 1954).Google Scholar
Bonet, J., Domański, P., Lindström, M. and Taskinen, J., ‘Composition operators between weighted Banach spaces of analytic functions’, J. Aust. Math. Soc. Ser. A 64 (1) (1998), 101–118.CrossRefGoogle Scholar
Bonet, J. and Vukotić, D., ‘Superposition operators between weighted Banach spaces of analytic functions of controlled growth’, Monatsh. Math. 170 (3–4) (2013), 311–323.Google Scholar
Bonet, J. and Wolf, E., ‘A note on weighted Banach spaces of holomorphic functions’, Arch. Math. 81 (2003), 650–654.CrossRefGoogle Scholar
Bourgain, J. and Talagrand, M., ‘Compacité extrémale’, Proc. Amer. Math. Soc. 80 (1) (1980), 68–70.CrossRefGoogle Scholar
Boyd, C. and Rueda, P., ‘Isometries of weighted spaces of holomorphic functions on unbounded domains’, Proc. R. Soc. Edinburgh 139A (2009), 253–271.CrossRefGoogle Scholar
Boyd, C. and Rueda, P., ‘The biduality problem and M-ideals in weighted spaces of holomorphic functions’, J. Convex Anal. 18 (4) (2011), 1065–1074.Google Scholar
Boyd, C. and Rueda, P., ‘Superposition operators between weighted spaces of analytic functions’, Quaest. Math. 36 (2013), 411–419.CrossRefGoogle Scholar
Buckley, S., Fernández, J. L. and Vukotić, D., Superposition operators on Dirichlet type spaces, in: Papers on Analysis, Rep. Univ. Jyväskylä Dep. Math. Stat., 83, (2001), 41–61.Google Scholar
Buckley, S. and Vukotić, D, ‘Univalent interpolation in Besov spaces and superposition into Bergman spaces’, Potential Anal. 29 (1) (2008), 1–16.Google Scholar
Cámera, G. A., ‘Nonlinear superposition on spaces of analytic functions’, in: Harmonic Analysis and Operator Theory (Caracas, 1994), Contemporary Mathematics, 189 (American Mathematical Society, Providence, RI, 1995), 103–116.CrossRefGoogle Scholar
Cámera, G. A. and Giménez, J., ‘The nonlinear superposition operator acting on Bergman spaces’, Compositio Math. 93 (1) (1994), 23–35.Google Scholar
Carando, D., García, D. and Maestre, M., ‘Homomorphisms and composition operators on algebras of analytic functions of bounded type’, Adv. Math. 197 (2) (2005), 607–629.Google Scholar
Cowen, C. C. and MacCluer, B. D., ‘Spectra of some composition operators’, J. Funct. Anal. 125 (1) (1994), 223–251.CrossRefGoogle Scholar
Cowen, C. C. and MacCluer, B. D., Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics (CRC Press, Boca Raton, FL, 1995).Google Scholar
Dineen, S., Complex Analysis on Infinite Dimensional Spaces, Monographs in Mathematics (Springer, London–Berlin–Heidelberg, 1999).CrossRefGoogle Scholar
Duren, P. and Schuster, A., Bergman Spaces, Mathematical Surveys and Monographs, 100 (American Mathematical Society, Providence, RI, 2004).CrossRefGoogle Scholar
García, D., Maestre, M. and Sevilla-Peris, P., ‘Weakly compact composition operators between weighted spaces’, Note Mat. 25 (1) (2005/06), 205–220.Google Scholar
Grosse-Erdmann, K.-G., ‘A weak criterion for vector-valued holomorphy’, Math. Proc. Cambridge Philos. Soc. 136 (2) (2004), 399–411.Google Scholar
Lindström, M. and Palmberg, N., ‘Duality of a large family of analytic function spaces’, Ann. Acad. Sci. Fenn. Math. 32 (1) (2007), 251–267.Google Scholar
Ryan, R. A., ‘Weakly compact holomorphic mappings on Banach spaces’, Pacific J. Math. 131 (1) (1988), 179–190.Google Scholar
Xiao, J., Holomorphic Q Classes, Lecture Notes in Mathematics, 1767 (Springer, Berlin–Heidelberg–New York, 2001).Google Scholar
Zhao, R., ‘On a general family of functions’, Ann. Acad. Sci. Fenn. Math. Diss. 105 (1996), 1–56.Google Scholar