Ces\`{a}ro-Type Operator on the Dirichlet Space of the Upper Half Plane
Abstract
We construct a Ces\`{a}ro-type operator acting on Dirichlet space of the upper half plane using the approach of strongly continuous semigroups of composition operators on Banach spaces. We then determine the spectral and norm properties of the obtained Ces\`{a}ro-type operator.
Keywords
Ces\`{a}ro-type operator, Generator, One-parameter semigroup, Resolvent, Spectrum
References
- [1] K. Zhu, Operator Theory in function spaces, Mathematical Surveys and Monographs, 138, American Mathematical Society, Providence, (2007).
- [2] O. El-Fallah, K. Kellay, J. Mashreghi, T. Ransford, A primer on the Dirichlet Space, Cambridge University Press, New York, 203, (2014).
- [3] M. M. Peloso, Classical Spaces of Holomorphic Functions Technical report, Universit di Milano, (2014).
- [4] J. M. Wanjiru, J. O. Bonyo, I. S. Wattanga, Reproducing Kernel for the Classical Dirichlet Space of the Upper Half Plane, IJFSCFRT, 15, (2022), 150-161.
- [5] K. Engel, R. Nagel, A Short Course on Operator Semigroups, Universitext, Springer, New York, 37, (2006).
- [6] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1970.
- [7] A. G. Siskakis, Semigroups of composition operators on the spaces of analytic functions, a review, Contemp. Math., 213, (1998), 229-252.
- [8] R. K. Singh, J. S. Manhas, Composition operators on functional spaces. Elsevier Science Publishers, Amsterdam, 179, 1993. s3-10(1), (1960), 376-394.
- [9] A. G. Siskakis, Semigroups of composition operators on Dirichlet spaces, Results Math., 30, (1996), 165-173.
- [10] A. G. Arvanitidis, A. G. Siskasis, Ces`aro Operators on the Hardy spaces of the half-plane, Canad. Math. Bull., 56(2), (2013), 229-240.
