## 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 $$B\in GL(2,\mathbb{C})$$.

### MSC:

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