Abstract
Abstract The aim of this paper is to characterize the solutions Φ : G → M 2(ℂ) of the following matrix functional equations Φ ( x y ) + 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 ( x y ) − 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.
Highlights
Throughout this paper, let G be a group with neutral element e, and σ : G → G be a homomorphism such that σ ◦ σ = id
We extend the setting of Φ from an abelian topological group to a group that need not be abelian
We show that any continuous solution of (1.1) on a compact group is abelian
Summary
The contribution of the present paper to the theory of matrix d’Alembert’s functional equations lies in the study of (1.1) on groups that need not be abelian. Our main contribution is a natural extension of the previous works [13, 14] to d’Alembert’s matrix functional equation. This knowledge in turn enables us to solve the symmetrized matrix multiplicative Cauchy equation (1.2). We show that any continuous solution of (1.1) on a compact group is abelian Another main result of this paper is the solution of the functional equation (1.4).
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