A spectral theorem for a non-Archimedean valued field whose residue field is formally real
Kosuke Ishizuka
TL;DR
The paper proves a spectral theorem for self-adjoint compactoid operators on non-Archimedean Banach spaces when the residue field $k$ is formally real, correcting earlier claims and clarifying precise hypotheses. It develops a Fredholm framework that does not require algebraic closedness, introduces the diagonalization condition $(H)_K$, and extends diagonalization results to Hahn-field extensions, showing the spectral data depend only on the residue field and are independent of the valuation group. The main result provides a decomposition $T(x)=\sum_{n=1}^\infty \lambda_n \frac{\langle x, x_n\rangle}{\langle x_n, x_n\rangle} x_n$ with norm relations that differ for densely versus discretely valued $K$, and establishes an equivalence between $(H)_K$ and $(H)_k$ to guarantee the spectral theorem under $(H)_k$. An appendix collects technical extensions to the compactoid-Fredholm theory without assuming $K$ is algebraically closed, including analytic function behavior and limit arguments in the Hahn-field setting.
Abstract
In this paper, we will prove a spectral theorem for self-adjoint compactoid operators. Also, we will study the condition on which the coefficient field must be imposed. In order to get the theorems, we will use the Fredholm theory for compactoid operators. Moreover, the property of maximal complete field is important for our study. These facts will allow us to find that the spectral theorem depends only on the residue class field, and is independent of the valuation group of the coefficient field. As a result, we can settle the problem of the spectral theorem in the case where the residue class field is formally real.