Berkovich spectra of elements in Banach Rings

Adapting the notion of the spectrum ${\mathrm{\Sigma }}_a$ for an element $a$ in an ultrametric Banach algebra (as defined by Berkovich), we introduce and briefly study the Berkovich spectrum ${\sigma }_R^{Ber}(u)$ 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 ${\sigma }_R^{Ber}(u)$ coincides with the spectrum of $R$ (as defined by Berkovich). If $R$ is a unital complex Banach algebra, then ${\sigma }_R^{Ber}(u)$ 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 ${\sigma }_{L(E)}^{Ber}(u)$. 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 $k$.