Berkovich spectra of elements in Banach Rings

TL;DR

This paper introduces and studies the Berkovich spectrum of elements in Banach rings, extending classical spectral notions to ultrametric and non-Archimedean contexts, with applications to Fredholm determinants.

Contribution

It defines the Berkovich spectrum for elements in Banach rings and explores its properties, unifying various spectral concepts in ultrametric and complex Banach algebras.

Findings

The Berkovich spectrum is a compact subset of the affine analytic space.

In complex Banach algebras, it relates to the usual spectrum via a folding process.

Provides bounds for zeros of Fredholm determinants in non-Archimedean fields.

Abstract

Adapting the notion of the spectrum $\Sigma_a$ for an element $a$ in an ultrametric Banach algebra (as defined by Berkovich), we introduce and briefly study the Berkovich spectrum $\sigma^{Ber}_R(u)$ of an element $u$ in a Banach ring $R$. This spectrum is a compact subset of the affine analytic space $A_Z^1$ over $Z$, and the later can be identified with the "equivalence classes" of all elements in all complete valuation fields. If $R$ is generated by $u$ as a unital Banach ring, then $\sigma^{Ber}_R(u)$ coincides with the spectrum of $R$ (as defined by Berkovich). If $R$ is a unital complex Banach algebra, then $\sigma^{Ber}_R(u)$ is the "folding up" of the usual spectrum $\sigma_B(u)$ alone the real axis. For a non-Archimedean complete valuation field $k$ and an infinite dimensional ultrametric $k$-Banach space $E$ with an orthogonal base, if $u\in L(E)$ is a completely continuous operator, we show that many different ways to define the spectrum of $u$ give the same compact set $\sigma^{Ber}_{L(E)}(u)$. As an application, we give a lower bound for the valuations of the zeros of the Fredholm determinant $\det(1- t\cdot u)$ (as defined by Serre) in complete valuation field extensions of $k$. Using this, we give a concrete example of a completely continuous operator whose Fredholm determinant does not have any zero in any complete valuation field extension of $k$.