On two notions of the local spectrum for several commuting operators (Q791118)

In order to study spectral decompositions and non-analytic functional calculi for systems \(a=(a_ 1,...,a_ n)\) of commuting Banach space operators E. Albrecht and S. Frunză deveolped two concepts of a local spectrum of a vector x with respect to a commuting system. Both local spectra are defined as the complement of the largest open set on which locally a certain equation can be solved. S. Frunză demanded the existence of a \(C^{\infty}\)-form f satisfying \(xs_ 1\wedge...\wedge s_ n=({\bar \partial}\oplus \alpha)f,\) while E. Albrecht demands vector valued analytic functions \(f_ 1,...,f_ n\) satisfying \(x=\sum^{n}_{i=1}(z_ i-a_ i)f_ i(z).\) Obviously the \(C^{\infty}\)-local spectrum is contained in the analytic local spectrum. The author used a standard procedure of homological algebra to prove the opposite inclusion thus showing the equivalence of both concepts. As an application a proof of S. Frunză for the uniqueness of the spectral capacity for commuting systems is replaced by a shorter and more elementary one.