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

Singh, Dinesh

Am. Math. Mon. 113, No. 8, 756-758 (2006).

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.

Reviewer: Dirk Werner (Berlin)

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

Keywords:

Banach algebra; spectrum