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A fixed-point theorem for holomorphic maps. (English) Zbl 0813.47068
Summary: We consider the action on the maximal ideal space \(M\) of the algebra \(H\) of bounded analytic functions, induced by an analytic self-map of a complex manifold \(X\). After some general preliminaries, we focus on the question of the existence of fixed points for this action, in the case when \(X\) is the open unit disk \(D\). We classify the fixed-point-free Möbius transformations, and we show that for an arbitrary analytic map from \(\mathbb{D}\) into itself, the induced map has a fixed point, or it restricts to a fixed-point-free Möbius map on some analytic disk contained in \(M\).
47H10 Fixed-point theorems
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces