<abstract xmlns="http://www.w3.org/1999/xhtml"><p>The aim of this paper is to characterize the solutions Φ : <italic>G → M</italic><sub>2</sub>(ℂ) of the following matrix functional equations
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2020-0025_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mfrac><mrow><mi mathvariant="normal">Φ</mi><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi mathvariant="normal">Φ</mi><mrow><mo>(</mo><mrow><mi>σ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></mfrac><mo>=</mo><mi mathvariant="normal">Φ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi mathvariant="normal">Φ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mo>∈</mo><mi>G</mi><mo>,</mo></mrow></math><tex-math>{{\Phi \left( {xy} \right) + \Phi \left( {\sigma \left( y \right)x} \right)} \over 2} = \Phi \left( x \right)\Phi \left( y \right),\,\,\,\,\,\,x,y, \in G,</tex-math></alternatives></disp-formula>
and
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2020-0025_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mfrac><mrow><mi mathvariant="normal">Φ</mi><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi></mrow><mo>)</mo></mrow><mo>−</mo><mi mathvariant="normal">Φ</mi><mrow><mo>(</mo><mrow><mi>σ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></mfrac><mo>=</mo><mi mathvariant="normal">Φ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi mathvariant="normal">Φ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mo>∈</mo><mi>G</mi><mo>,</mo></mrow></math><tex-math>{{\Phi \left( {xy} \right) - \Phi \left( {\sigma \left( y \right)x} \right)} \over 2} = \Phi \left( x \right)\Phi \left( y \right),\,\,\,\,\,\,x,y, \in G,</tex-math></alternatives></disp-formula>
where <italic>G</italic> is a group that need not be abelian, and <italic>σ</italic> : <italic>G → G</italic> is an involutive automorphism of <italic>G</italic>. Our considerations are inspired by the papers [13, 14] in which the continuous solutions of the first equation on abelian topological groups were determined.</p></abstract>

The aim of this paper is to characterize the solutions Φ : G → M2(ℂ) of the following matrix functional equations
Φ(xy)+Φ(σ(y)x)2=Φ(x)Φ(y),      x,y,∈G,{{\Phi \left( {xy} \right) + \Phi \left( {\sigma \left( y \right)x} \right)} \over 2} = \Phi \left( x \right)\Phi \left( y \right),\,\,\,\,\,\,x,y, \in G,
and
Φ(xy)−Φ(σ(y)x)2=Φ(x)Φ(y),      x,y,∈G,{{\Phi \left( {xy} \right) - \Phi \left( {\sigma \left( y \right)x} \right)} \over 2} = \Phi \left( x \right)\Phi \left( y \right),\,\,\,\,\,\,x,y, \in 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.

A Variant of D’alembert’s Matrix Functional Equation

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"A Variant of D\u2019alembert\u2019s Matrix Functional Equation"}

The aim of this paper is to characterize the solutions Φ : G → M2(ℂ) of the following matrix functional equations
Φ(xy)+Φ(σ(y)x)2=Φ(x)Φ(y),      x,y,∈G,{{\Phi...

A Variant of D’alembert’s Matrix Functional Equation

Publié en ligne: 14 déc. 2020

Pages: 21 - 43

Reçu: 15 juil. 2020

Accepté: 19 nov. 2020

DOI: https://doi.org/10.2478/amsil-2020-0025

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

© 2020 Youssef Aissi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

A Variant of D’alembert’s Matrix Functional Equation

Publié en ligne: 14 déc. 2020

Pages: 21 - 43

Reçu: 15 juil. 2020

Accepté: 19 nov. 2020

DOI: https://doi.org/10.2478/amsil-2020-0025

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

© 2020 Youssef Aissi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

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