Möbius’s functional equation and Schur’s lemma with applications to the complex unit disk

Abstract

Mobius addition is defined on the complex open unit disk by $$\begin{aligned} a\oplus _M b = \dfrac{a+b}{1+\bar{a}b} \end{aligned}$$ and Mobius’s exponential equation takes the form $$L(a\oplus _M b) = L(a)L(b)$$ , where L is a complex-valued function defined on the complex unit disk. In the present article, we indicate how Mobius’s exponential equation is connected to Cauchy’s exponential equation. Mobius’s exponential equation arises when one determines the irreducible linear representations of the unit disk equipped with Mobius addition, considered as a nonassociative group-like structure. This suggests studying Schur’s lemma in a more general setting.