Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access May 16, 2016

Invertible and normal composition operators on the Hilbert Hardy space of a half–plane

Valentin Matache
From the journal Concrete Operators

Abstract

Operators on function spaces of form Cɸf = f ∘ ɸ, where ɸ is a fixed map are called composition operators with symbol ɸ. We study such operators acting on the Hilbert Hardy space over the right half-plane and characterize the situations when they are invertible, Fredholm, unitary, and Hermitian. We determine the normal composition operators with inner, respectively with Möbius symbol. In select cases, we calculate their spectra, essential spectra, and numerical ranges.

References

[1] Bourdon P. S., Matache V., Shapiro J. H., On convergence to the Denjoy-Wolff point, Illinois J. Math. 49 (2005), no. 2, 405-430. Search in Google Scholar

[2] Bourdon P. S., Narayan S. K., Normal weighted composition operators on the Hardy space H2.U/, J. Math. Anal. Appl. 367(2010), 278-286. 10.1016/j.jmaa.2010.01.006Search in Google Scholar

[3] Cowen, C. C., Ko, E., Hermitian weighted composition operators on H2, Trans. Amer. Math. Soc., 362(2010), no. 11, 5771-5801. Search in Google Scholar

[4] Duren P., Theory of Hp Spaces, Pure and Applied Mathematics, Vol. 38 Academic Press, New York–London 1970. Search in Google Scholar

[5] Elliott S., Jury M. T., Composition operators on Hardy spaces of a half-plane, Bull. Lond. Math. Soc. 44 (2012), no. 3, 489-495. Search in Google Scholar

[6] Gunatillake G., Invertible weighted composition operators, J. Funct. Anal. 261 (2011), no. 3, 831-860. Search in Google Scholar

[7] Hoffman K., Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1962. Search in Google Scholar

[8] Hyvarinen O., Lindstrom M., Nieminen I., Saukko E., Spectra of weighted composition operators with automorphic symbols, J. Funct. Anal. 265 (2013), 1749-1777. 10.1016/j.jfa.2013.06.003Search in Google Scholar

[9] Matache V., Composition operators on Hp of the upper half-plane, An. Univ. Timis¸oara Ser. S¸ tiin¸t. Mat. 27 (1989), no. 1, 63-66. Search in Google Scholar

[10] Matache V., Notes on hypercyclic operators, Acta Sci. Math. (Széged) 58(1993), no. 1-4, 401-410. Search in Google Scholar

[11] Matache V., Composition operators on Hardy spaces of a half-plane, Proc. Amer. Math. Soc. 127 (1999), no. 5, 1483-1491. Search in Google Scholar

[12] Matache V., Weighted composition operators on H2 and applications, Complex Anal. Oper. Theory 2 (2008), no. 1, 169-197. Search in Google Scholar

[13] Matache V., Numerical ranges of composition operators with inner symbols, Rocky Mountain J. Math. 42(2012), no. 1, 235-249. Search in Google Scholar

[14] Matache V., Isometric weighted composition operators, New York J. Math. 20(2014), 711-726. Search in Google Scholar

[15] Nordgren E. A., Composition operators, Canad. J. Math. 20(1968), 442-449. 10.4153/CJM-1968-040-4Search in Google Scholar

Received: 2015-10-9
Accepted: 2016-4-27
Published Online: 2016-5-16

© 2016 Valentin Matache

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 10.12.2022 from https://www.degruyter.com/document/doi/10.1515/conop-2016-0009/html
Scroll Up Arrow