A variant of d’Alembert’s matrix functional equation

Youssef Aissi; Driss Zeglami; Mohamed Ayoubi

Published: 2020-12-14

Vol. 35 No. 1 (2021)

A variant of d’Alembert’s matrix functional equation

Youssef Aissi , Driss Zeglami

, Mohamed Ayoubi

Abstract

The aim of this paper is to characterize the solutions Φ:G→M₂(ℂ) of the following matrix functional equations
\frac{Φ(xy)+Φ(σ(y)x)}{2} = Φ(x)Φ(y), x,y∈G,
and
\frac{Φ(xy)-Φ(σ(y)x)}{2} = Φ(x)Φ(y), x,y∈G,
where G is a group that need not be abelian, and σ:G→G is an involutive automorphism of G. Our considerations are inspired by the papers [13, 14] in which the continuous solutions of the first equation on abelian topological groups were determined.

Keywords:

matrix functional equation , d’Alembert , character , quadratic equation , morphism , symmetrized additive Cauchy equation , linear algebra

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Aissi, Y., Zeglami, D., & Ayoubi, M. (2020). A variant of d’Alembert’s matrix functional equation. Annales Mathematicae Silesianae, 35(1), 21–43. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13472

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Youssef Aissi

Affiliation:

Department of Mathematics, E.N.S.A.M, Moulay ISMAÏL University, Morocco Morocco

Driss Zeglami

zeglamidriss@yahoo.fr
https://orcid.org/0000-0001-7493-5931

Affiliation:

Department of Mathematics, E.N.S.A.M, Moulay ISMAÏL University, Morocco Morocco

Mohamed Ayoubi

Affiliation:

Department of Mathematics, E.N.S.A.M, Moulay ISMAÏL University, Morocco Morocco

Vol. 35 No. 1 (2021)
Published: 2021-02-10

ISSN: 0860-2107

eISSN: 2391-4238

10.2478/amsil

Publisher

University of Silesia Press

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