D’Alembert’s and Wilson’s functional equations for vector and \(2\times 2\) matrix valued functions. (English) Zbl 0968.39012

The author considers D’Alembert and Wilson’s functional equations on Abelian groups with values in \(\mathbb{C}^2\) or \(M_2(\mathbb{C})\), the algebra of \(2\times 2\) matrices over \(\mathbb{C}\). Among others, he proves the following result: Let \(G\) be an Abelian topological group, \(\sigma:G\to G\) be a continuous homomorphism such that \(\sigma^2=I\), \(M(G)\) be the set of all continuous homomorphisms \(\gamma:G \to\mathbb{C}^*\). Let \(\Phi: G\to M_2(\mathbb{C})\) be continuous and satisfy the functional equation \[ \Phi(x+y) +\Phi(x+ \sigma y)=2 \Phi(y) \Phi(x). \tag{1} \] Result. The continuous solutions \(\Phi\) of (1) with \(\Phi(0)=I\) are given by \[ \Phi=B \left(\begin{matrix} (\gamma_1+ \gamma_2 \circ \sigma)/2 & 0\\ 0 & (\gamma_2+ \gamma_2\circ \sigma)/2\end{matrix} \right)B^{-1} \] where \(\gamma_1, \gamma_2\in {\mathcal M}(G)\). \[ \Phi=B \left(\begin{matrix} (\gamma+ \gamma \circ \sigma)/2 & (\gamma+ \gamma\circ \sigma)a^+/2 +(\gamma-\gamma \circ \sigma) a^-/2\\ 0 & (\gamma+\gamma \circ\sigma)/2\end{matrix}\right)B^{-1} \] where \(\gamma\in {\mathcal M}(G)\) has \(\gamma\neq \gamma \circ \sigma\) and where \(a^\pm\in {\mathcal A}^\pm (G)=\) the set of all additives continuous \(a:G\to \mathbb{C}\) such that \(a\circ\sigma =\pm a\), \[ \Phi=B \gamma^+ \left(\begin{matrix} 1 & a^++S^-\\ 0 & 1 \end{matrix} \right)B^{-1} \] where \(B\in GL(2,\mathbb{C})\).


39B42 Matrix and operator functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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