An operator on some ultrametric Hilbert spaces
type: Item
instance of: scholarly article
title: An operator on some ultrametric Hilbert spaces
zbMATH Open document ID: 0930.47049
author: Bertin Diarra
published in: The Journal of Analysis
publication date: 1999-08-25T00:00:00Z
review text: Let \(K\) be a complete valued field with valuation \(v\) and value group \(\Gamma\). Let \(\omega= (\omega_i)_{i\geq 0}\subset K\setminus\{0\}\) and let \(E_\omega\) be the Hilbert space associated with \(\omega\), i.e. the space of sequences \(x= (x_i)_{i\geq 0}\subset K\) such that \(\lim_{i\to\infty} x^2_i\omega_i= 0\). In the case where \(\omega_0= 1\), \(\lim_{i\to\infty} \omega^{-1}_i= 0\) and the sequence \((v(\omega_j))_{j\geq 0}\) is strictly decreasing in \(\Gamma\), one can define, as done by \textit{H. A. Keller} and \textit{H. A. Ochsenius} [Math. Slovaca 45, No. 4, 413-434 (1995; Zbl 0855.46049)], for a specific sequence \(\omega\) in a specific valued field \(K\), the operator \(A\) on \(E_\omega\) by setting on the canonical orthogonal base \((e_i)_{i\geq 0}\) of \(E_\omega\), \(Ae_i= e_i+ \sum_{j\neq i} \omega^{-1}_j e_j\). In the specific example studied by Keller and Ochsenius, the value group is \(\Gamma= \mathbb{Z}^{(\mathbb{N})}\) with the antilexicographic order. They showed that the operator \(A\) is topologically irreducible and \(\text{spec}(A)= \{1\}\). In this paper, the author shows that in the rank one value case the situation is different. In fact, the operator \(A\) has eigenvalues and its eigenvalues are of the form \(\lambda= 1+\alpha\), where \(\alpha\) runs over the zeros of the function \(\phi(\alpha)= 1-\sum_{j\geq 0}{1\over 1+\alpha\omega_j}\). The zeros of \(\phi\) are located by classical \(p\)-adic analytic function theory. Moreover, it is shown that \(\text{spec}(A)= \{1\}\cup\{\text{eigenvalues of }A\}\) and \(1\) is the limit of the sequence of eigenvalues.
Mathematics Subject Classification ID: 47S10
Mathematics Subject Classification ID: 47A75
zbMATH DE Number: 1269667
zbMATH Keywords: valued field
zbMATH Keywords: conical orthogonal base
zbMATH Keywords: antilexicographic order
zbMATH Keywords: eigenvalues
zbMATH Keywords: valuation
zbMATH Keywords: value group
zbMATH Keywords: \(p\)-adic analytic function theory
reviewed by: T. D. Narang
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1269667