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Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions

Published online by Cambridge University Press:  20 November 2018

José Bonet ,
Paweł Dománski  and
Mikael Lindström
José Bonet
Affiliation:
Departamento de Matemática Aplicada E. T. S. Arquitectura Universidad Politécnica de Valencia E-46071 Valencia Spain
Paweł Dománski
Affiliation:
Department of Mathematics Åbo Akademi University FIN-20500 Åbo Finland
Mikael Lindström
Affiliation:
Faculty of Mathematics and Computer Science A. Mickiewicz University Ul. Matejki 48/49 60-769 Poznán Poland
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Abstract

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Every weakly compact composition operator between weighted Banach spaces $H_{v}^{\infty }$ of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space $H_{v}^{\infty }$ are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces.

Keywords

47B3830D5546E15weighted Banach spaces of holomorphic functionscomposition operatorcompact operatorweakly compact operator
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 42 , Issue 2 , 01 June 1999 , pp. 139 - 148
Copyright
Copyright © Canadian Mathematical Society 1999

References

[1] Anderson, J. M. and Duncan, J., Duals of Banach spaces of entire functions. Glasgow Math. J. 32 (1990), 215–220.Google Scholar
[2] Aron, R., Galindo, P. and Lindström, M., Compact homomorphisms between algebras of analytic functions. Studia Math. 123 (1997), 235–247.Google Scholar
[3] Berenstein, C. A. and Gay, F., Complex Variables, An Introduction. Springer, 1991.Google Scholar
[4] Bierstedt, K. D., Bonet, J. and Galbis, A.,Weighted spaces of holomorphic functions on bounded domains. MichiganMath. J. 40 (1993), 271–297.Google Scholar
[5] Bierstedt, K. D., Bonet, J. and Taskinen, J., Associated weights and spaces of holomorphic functions. StudiaMath. 127 (1998), 137–168.Google Scholar
[6] Bierstedt, K. D. and Summers, W. H., Biduals of weighted Banach spaces of analytic functions. J. Austral. Math. Soc. (Series A) 54 (1993), 70–79.Google Scholar
[7] Bonet, J., Dománski, P., Lindström, M. and Taskinen, J., Composition operators between weighted Banach spaces of analytic functions. J. Austral.Math. Soc. 64 (1998), 101–118.Google Scholar
[8] Conway, J., Functions of One Complex Variable. Springer, 1978.Google Scholar
[9] Cowen, C. and MacCluer, B., Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton, 1995.Google Scholar
[10] Diestel, J., Sequences and series in Banach spaces. Springer, 1984.Google Scholar
[11] Galbis, A., Weighted Banach spaces of entire functions. Arch. Math. 62 (1994), 58–64.Google Scholar
[12] Hoffman, K., Banach spaces of analytic functions. Prentice Hall, Englewood Cliffs, 1967.Google Scholar
[13] Jarchow, H., Some functional analytic properties of composition operators. Quaestiones Math. 18 (1995), 229–256.Google Scholar
[14] Kaballo, W., Lifting-Probleme für H1-Funktionen. Arch. Math. 34 (1980), 540–549.Google Scholar
[15] Liu, P., Saksman, E. and Tylli, H.-O., Small composition operators on analytic vector-valued function spaces. Pacific J. Math. (to appear).Google Scholar
[16] Lusky, W., On the structure of Hv0(D) and hv0(D). Math. Nachr. 159 (1992), 279–289.Google Scholar
[17] Lusky, W., On weighted spaces of harmonic and holomorphic functions. J. London Math. Soc. 51 (1995), 309–320.Google Scholar
[18] Momm, S., Ideale in gewichteten Algebren holomorpher Funktionen auf dem Einheits-kreis. Dissertation, Düsseldorf, 1988.Google Scholar
[19] Montes-Rodrigues, A., The essential norm of a composition operator on Bloch spaces. Pacific J. Math., to appear.Google Scholar
[20] Poggi-Corradini, P., Hardy spaces and twisted sectors for geometric models. Trans. Amer. Math. Soc. 348 (1996), 2503–2518.Google Scholar
[21] Rosenthal, H. P., On relatively disjoint families of measures with some applications to Banach space theory. Studia Math. 37 (1970), 13–36.Google Scholar
[22] Rubel, L. A. and Shields, A. L., The second duals of certain spaces of analytic functions. J. Austral. Math. Soc. 11 (1970), 276–280.Google Scholar
[23] Shapiro, J. H., Composition Operators and Classical Function Theory. Springer, 1993.Google Scholar
[24] Shapiro, J. H., The essential norm of a composition operator. Ann.Math. 125 (1987), 375–404.Google Scholar
[25] Shields, A. L. and Williams, D. L., Bounded projections, duality, and multipliers in spaces of harmonic functions. J. Reine Angew.Math. 299/300 (1978), 256–279.Google Scholar
[26] Shields, A. L. and Williams, D. L., Bounded projections and the growth of harmonic conjugates in the disc. Michigan Math. J. 29 (1982), 3–25.Google Scholar
[27] Taskinen, J., Compact composition operators on general weighted spaces. Preprint, 1996.Google Scholar
[28] ¨Ulger, A., Some recent results about the spectrum of commutative Banach algebras under the weak topology and applications. Monatsh.Math. 121 (1996), 353–379.Google Scholar
[29] Wojtaszczyk, P., Banach spaces for Analysts. Cambridge, 1991.Google Scholar