$C$-groups and mixed $C$-groups of bounded linear operators on non-archimedean Banach spaces

Authors

  • Abdelkhalek El Amrani Department of mathematics and computer science, Sidi Mohamed Ben Abdellah University, Faculty of Sciences Dhar El Mahraz, Atlas F`es, Morocco
  • Aziz Blali Ecole Normale Sup´erieure, Department of mathematics, Sidi Mohamed Ben Abdellah University, Bensouda-F`es, Morocco
  • Jawad Ettayb Department of mathematics and computer science, Sidi Mohamed Ben Abdellah University, Faculty of Sciences Dhar El Mahraz, Atlas F`es, Morocco

DOI:

https://doi.org/10.33044/revuma.2074

Abstract

We introduce and study $C$-groups and mixed $C$-groups of bounded linear operators on non-archimedean Banach spaces. Our main result extends some existing theorems on this topic. In contrast with the classical setting, the parameter of a given $C$-group (or mixed $C$-group) belongs to a clopen ball $\Omega_{r}$ of the ground field $\mathbb{K}$. As an illustration, we discuss the solvability of some homogeneous $p$-adic differential equations for $C$-groups and inhomogeneous $p$-adic differential equations for mixed $C$-groups when $\alpha=-1$. Examples are given to support our work.

Downloads

Download data is not yet available.

References

R. deLaubenfels, $C$-semigroups and strongly continuous semigroups, Israel J. Math. 81 (1993), no. 1-2, 227–255. MR 1231189.

T. Diagana, $C_0$-semigroups of linear operators on some ultrametric Banach spaces, Int. J. Math. Math. Sci. 2006, Art. ID 52398, 9 pp.

T. Diagana and F. Ramaroson, Non-Archimedean Operator Theory, SpringerBriefs in Mathematics, Springer, 2016. MR 3468878.

B. Diarra, An operator on some ultrametric Hilbert spaces, J. Anal. 6 (1998), 55–74. MR 1671148.

A. El Amrani, A. Blali, J. Ettayb and M. Babahmed, A note on $C_0$-groups and $C$-groups on non-archimedean Banach spaces, Asian-Eur. J. Math. 14 (2021), no. 6, Paper No. 2150104, 19 pp. MR 4280924.

N. Koblitz, $p$-adic Analysis: a Short Course on Recent Work, London Mathematical Society Lecture Note Series, 46, Cambridge University Press, 1980. MR 0591682.

M. Mosallanezhad and M. Janfada, On mixed $C$-semigroups of operators on Banach spaces, Filomat 30 (2016), no. 10, 2673–2682. MR 3583393.

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, 1983. MR 0710486.

C. Perez-Garcia and W. H. Schikhof, Locally Convex Spaces Over Non-Archimedean Valued Fields, Cambridge Studies in Advanced Mathematics, 119, Cambridge University Press, 2010. MR 2598517.

A. C. M. van Rooij, Non-Archimedean Functional Analysis, Monographs and Textbooks in Pure and Applied Mathematics, 51, Marcel Dekker, 1978. MR 0512894.

Downloads

Published

2022-05-30

Issue

Section

Article