Harmonic functional calculi in Banach algebras with involution

Abstract

Harmonic functional calculi were defined separately by Akkar, El Kinani and Oudadess, and by Foias. In the first paper, the authors used the Poisson integral formula to define a functional calculus. In the second paper, the author used the fact that real harmonic functions on simply connected domains are real parts of analytic functions, then, he defined a harmonic calculus from the analytic one. We will present these two functional calculi and prove that they are equivalent in the sense of the Rhinehart.