Nonarchimedean spectral theory
type: Item
instance of: scholarly article
title: Nonarchimedean spectral theory
zbMATH Open document ID: 0571.46052
DOI: 10.1007/BF02249122
published in: Journal of Soviet Mathematics
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46S10
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47B40
zbMATH DE Number: 3911077
zbMATH Keywords: non-Archimedean valued field
zbMATH Keywords: analytic operator with compact spectrum
zbMATH Keywords: spectral theorem
zbMATH Keywords: functional calculus
zbMATH Keywords: non-Archimedean distributions
zbMATH Keywords: completely continuous operators
zbMATH Keywords: operators of scalar type
zbMATH Keywords: normal operators
zbMATH Keywords: non-Archimedean zeta-function
author: Misha Vishik
reviewed by: Willy J. F. Govaerts
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3911077