A variant of d’Alembert’s matrix functional equation
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
References
2. J.A. Baker and K.R. Davison, Cosine, exponential and quadratic functions, Glasnik Mat. Ser. III 16(36) (1981), no. 2, 269–274.
3. W. Chojnacki, Fonctions cosinus hilbertiennes bornées dans les groupes commutatifs localement compacts, Compositio Math. 57 (1986), no. 1, 15–60.
4. N. Dunford and J.T. Schwartz, Linear Operators. Part 1: General Theory, Interscience Publishers, Inc., New York, 1958.
5. B. Fadli, D. Zeglami, and S. Kabbaj, A variant of Wilson’s functional equation, Publ. Math. Debrecen 87 (2015), no. 3-4, 415–427.
6. B. Fadli, D. Zeglami, and S. Kabbaj, A variant of the quadratic functional equation on semigroups, Proyecciones 37 (2018), no. 1, 45–55.
7. H.O. Fattorini, Uniformly bounded cosine functions in Hilbert space, Indiana Univ. Math. J. 20 (1970/71), 411–425.
8. J. Kisyński, On operator-valued solutions of d’Alembert’s functional equation. I, Colloq. Math. 23 (1971), 107–114.
9. S. Kurepa, Uniformly bounded cosine function in a Banach space, Math. Balkanica 2 (1972), 109–115.
10. H. Radjavi and P. Rosenthal, Simultaneous Triangularization, Springer-Verlag, New York, 2000.
11. KH. Sabour, B. Fadli, and S. Kabbaj,Trigonometric functional equations on monoids, Asia Mathematika 3 (2019), no. 1, 1–9.
12. P. Sinopoulos, Wilson’s functional equation for vector and matrix functions, Proc. Amer. Math. Soc. 125 (1997), no. 4, 1089–1094.
13. H. Stetkær, Functional equations on abelian groups with involution, Aequationes Math. 54 (1997), no. 1–2, 144–172.
14. H. Stetkær, D’Alembert’s and Wilson’s functional equations for vector and 2x2 matrix valued functions, Math. Scand 87 (2000), no. 1, 115–132.
15. H. Stetkær, Functional Equations on Groups, World Scientific Publishing Co., Singapore, 2013.
16. H. Stetkær, A variant of d’Alembert’s functional equation, Aequationes Math. 89 (2015), no. 3, 657–662.
17. L. Székelyhidi, Functional equations on abelian groups, Acta Math. Acad. Sci. Hungar. 37 (1981), no. 1–3, 235–243.
Department of Mathematics, E.N.S.A.M, Moulay ISMAÏL University, Morocco Morocco
Department of Mathematics, E.N.S.A.M, Moulay ISMAÏL University, Morocco Morocco
https://orcid.org/0000-0001-7493-5931
Department of Mathematics, E.N.S.A.M, Moulay ISMAÏL University, Morocco Morocco
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