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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