On norms of composition operators on weighted hardy spaces
DOI:
https://doi.org/10.24297/jam.v14i1.6717Abstract
The computation of composition operator on Hardy spaces is very hard. In this paper we propose a norm of a bounded composition operator on weighted Hardy spaces H2(b) induced by a disc automorphism by embedding the classical Hardy space . The estimate obtained is accurate in the sense that it provides the exact norm for particular instances of the sequence b. As an application of our results, an estimate for the norm of any bounded composition operator on H2(b) is obtained.
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References
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[2] Axler S., Jewell N. and Shields A., 1980, The essential norm of an operator and its adjoint, Trans. Amer. Math. Soc. 261 (1980), 159–167.
[3] Appel M. J., Bourdon P. S. and Thrall J. J., 1996, Norms of composition operators in the Hardy space, Experiment. Math. 5 (1996), 111–117.
[4] Berkson E. and Porta H.,1978, Semigroups of analytic functions and composition operators, Michigan Math. J., 25 (1978), no. 1, 101–115.
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[11] Gallardo-Gutierrez E. A. and Montes-Rodrıguez A.,2004 The role of the spectrum in the cyclic behavior of composition operators, Mem. Amer. Math. Soc. 167 (2004), no. 791, x+81 pp.
[12] Hammond C.,2003, On the norm of a composition operator with linear fractional symbol, Acta Sci. Math. (Szeged) 69 (2003), no. 3-4, 813–82
[13] Hammond C.,2006, Zeros of hypergeometric functions and the norm of a composition operator Comput. Methods Funct. Theory 6 (2006), no. 1, 37–50.
[14] Hammond C. and Patton L. J.,2010, Norm inequalities for composition operators on Hardy and weighted Bergman spaces, Topics in operator theory. Volume 1. Operators, matrices and analytic functions, 265–272, Oper. Theory Adv. Appl., 202, Birkhauser Verlag, Basel, 2010.
[15] Kriete T. L. and . MacCluer B. D,1992, Composition operators on large weighted Bergman spaces, Indiana Univ. Math. J. 41 (1992), no. 3, 755–788.
[16] MacCluer B. D., Zeng X. and Zorboska N.,1996, Composition operators on small weighted Hardy spaces, Illinois J. Math. 40 (1996), no. 4, 662–677.
[17] . Shapiro J. H,1993, Composition operators and classical function theory, Springer-Verlag, 1993.
[18] Shields A. L.,1974, Weighted shift operators and analytic function theory, Topics in Operator Theory, Math. Surveys Monographs, Amer. Math. Soc., Providence, RI 13 (1974), 49-128.
[19] Zorboska N.,1990, Composition operators on Sa spaces, Indiana Univ. Math. J. 39 (1990), 847-857.
[2] Axler S., Jewell N. and Shields A., 1980, The essential norm of an operator and its adjoint, Trans. Amer. Math. Soc. 261 (1980), 159–167.
[3] Appel M. J., Bourdon P. S. and Thrall J. J., 1996, Norms of composition operators in the Hardy space, Experiment. Math. 5 (1996), 111–117.
[4] Berkson E. and Porta H.,1978, Semigroups of analytic functions and composition operators, Michigan Math. J., 25 (1978), no. 1, 101–115.
[5] Bourdon P. S., Fry E. E., Hammond C. and Spofford C. H., 2004, Norms of linear-fractional composition operators, Trans. Amer. Math. Soc. 356 (2004), 2459–2480.
[6] Cowen C. C.,1983, Composition operators on H2, J. Operator Theory, 9 (1983), 77–106
[7] Cowen C. C.,1990, An application of Hadamard multiplication to operators on weighted Hardy spaces, Linear Alg. Appl. 133 (1990), 21–32.
[8] Cowen C. C. and MacCluer B. D.,1995, Composition Operators on Spaces of Analytic Functions, CRC Press, 1995.
[9] Davies E. B.,2007, Linear operators and their spectra, Cambridge University Press, 2007.
[10] Galindo P and . Lindstrom M,2010, Essential norm of operators on weighted Bergman spaces of inï¬nite order, J. Oper. Theor. 64 (2010), 387–399.
[11] Gallardo-Gutierrez E. A. and Montes-Rodrıguez A.,2004 The role of the spectrum in the cyclic behavior of composition operators, Mem. Amer. Math. Soc. 167 (2004), no. 791, x+81 pp.
[12] Hammond C.,2003, On the norm of a composition operator with linear fractional symbol, Acta Sci. Math. (Szeged) 69 (2003), no. 3-4, 813–82
[13] Hammond C.,2006, Zeros of hypergeometric functions and the norm of a composition operator Comput. Methods Funct. Theory 6 (2006), no. 1, 37–50.
[14] Hammond C. and Patton L. J.,2010, Norm inequalities for composition operators on Hardy and weighted Bergman spaces, Topics in operator theory. Volume 1. Operators, matrices and analytic functions, 265–272, Oper. Theory Adv. Appl., 202, Birkhauser Verlag, Basel, 2010.
[15] Kriete T. L. and . MacCluer B. D,1992, Composition operators on large weighted Bergman spaces, Indiana Univ. Math. J. 41 (1992), no. 3, 755–788.
[16] MacCluer B. D., Zeng X. and Zorboska N.,1996, Composition operators on small weighted Hardy spaces, Illinois J. Math. 40 (1996), no. 4, 662–677.
[17] . Shapiro J. H,1993, Composition operators and classical function theory, Springer-Verlag, 1993.
[18] Shields A. L.,1974, Weighted shift operators and analytic function theory, Topics in Operator Theory, Math. Surveys Monographs, Amer. Math. Soc., Providence, RI 13 (1974), 49-128.
[19] Zorboska N.,1990, Composition operators on Sa spaces, Indiana Univ. Math. J. 39 (1990), 847-857.
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Published
2018-01-11
How to Cite
Shammaky, A. E., & Dalal, S. (2018). On norms of composition operators on weighted hardy spaces. JOURNAL OF ADVANCES IN MATHEMATICS, 14(1), 7424–7430. https://doi.org/10.24297/jam.v14i1.6717
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