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Calculs fonctionnels harmonique et analytique réel. (Real analytic and harmonic functional calculi). (French. English summary) Zbl 0673.46029
Summary: We define and study two new functional calculi. The first one, say harmonic, in Banach algebras with continuous involution, consists in giving a sense to \(f(x)\) whenever \(x\) is an element of such algebra and \(f\) is a harmonic function in an open set \(U\) containing Sp \(x\). The second, say real analytic, in hermitian commutative Banach algebras, consists in giving a sense to \(f(x)\) whenever \(x\) is an element of such algebra and \(f\) is an analytic function in two real variables on a neighborhood \(U\) of Sp \(x\).

MSC:
46H30 Functional calculus in topological algebras
47A60 Functional calculus for linear operators
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