Nonarchimedean spectral theory
Let C be a complete nontrivially non-Archimedean valued field. Let J be any index set and \(B=C(J)\) the space of families \((a_ j)_{j\in J}\) which are finite at infinity, provided with the supremum norm. An operator \({\mathcal A}\in End(B)\) is called an analytic operator with compact spectrum if 1) the spectrum \(\sigma_{{\mathcal A}}\) is compact 2) the resolvent is analytic on the complement of \(\sigma_{{\mathcal A}}\), i.e. the map \(z\to <h,(z-{\mathcal A})^{-1}g>\) is analytic there in the sense of Krasner for all \(g\in B\) and \(h\in B^*.\) A spectral theorem is proved for such operators so that functional calculus becomes feasible. Among other interesting results (e.g. on non-Archimedean distributions) it is proved that the analytic operators include all completely continuous operators. They also include the operators of scalar type which are analogous to classical normal operators on Hilbert space and which are related to the non-Archimedean zeta-function. The paper is carefully written and so rather readable in spite of its comprehensive nature. It also gives good connections with previous work of Dwork, Krasner, Serre and others.
scientific article; zbMATH DE number 3911077
scientific article