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The spectrum in a Banach algebra. (English) Zbl 1146.46022

The author presents a proof of the fact that the spectrum of every element in a unital Banach algebra is non-empty based on “differentiation under the integral” techniques, thus avoiding the use of complex analysis methods. The same type of reasoning enables him to give a real-variable proof of Liouville’s theorem on bounded entire functions.

MSC:

46H05 General theory of topological algebras
46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
47A10 Spectrum, resolvent
30D10 Representations of entire functions of one complex variable by series and integrals
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