Abstract
Let \(\alpha\) be a complex scalar, and let \(A\) be a bounded linear operator on a Hilbert space \(H\). We say that \(\alpha\) is an extended eigenvalue of \(A\) if there exists a nonzero bounded linear operator \(X\) such that \(AX=\alpha XA\). In weighted Hardy spaces invariant under automorphisms, we completely compute the extended eigenvalues of composition operators induced by linear fractional self-mappings of the unit disk \(\mathbb{D}\) with one fixed point in \(\mathbb{D}\) and one outside \(\overline{\mathbb{D}}\). Such classes of transformations include elliptic and loxodromic mappings as well as a hyperbolic nonautomorphic mapping.
Similar content being viewed by others
References
C. Cowen and B. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, FL, 1995.
M. Lacruz, F. León-Saavedra, S. Petrovic, and L. Rodríguez-Piazza, “Extended eigenvalues for composition operators”, J. Math. Anal. Appl., 504:2 (2021).
P. Lefèvre, D. Li, H. Queffélec, and L. Rodríguez-Piazza, Boundedness of composition operators on general weighted Hardy spaces of analytic functions, arXiv: 2011.14928.
J. E. Littlewood, “On inequalities in the theory of functions”, Proc. London Math. Soc. (2), 23:7 (1925), 481–519.
N. Zorboska, Composition operators on weighted Hardy spaces, University of Toronto, Toronto, 1988.
N. Zorboska, “Compact composition operators on some weighted Hardy spaces”, J. Operator Theory, 22:2 (1989), 233–241.
N. Zorboska, “Angular derivative and compactness of composition operators on large weighted Hardy spaces”, Canad. Math. Bull., 37:3 (1994), 428–432.
Acknowledgments
The first author was supported by Aula Universitaria del Estrecho, Plan Propio UCA-Internacional. The second and the third authors were supported in part by Ministerio de Ciencia, Innovación y Universidades (Spain), grant PGC2018-101514-B-I00 and by the 2014-2020 ERDF Operational Programme and by the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia. Project reference: FEDER-UCA18-108415.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2022, Vol. 56, pp. 3–9 https://doi.org/10.4213/faa3906.
Rights and permissions
About this article
Cite this article
Bensaid, I., León-Saavedra, F. & Romero de la Rosa, P. Extended Spectra for Some Composition Operators on Weighted Hardy Spaces. Funct Anal Its Appl 56, 81–85 (2022). https://doi.org/10.1134/S0016266322020010
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0016266322020010