Spectral Theory for $p$-adic Operators

TL;DR

This paper develops an algorithm to determine when a matrix over a local field can be diagonalized by a unitary matrix, introducing the notion of normality for $p$-adic operators and exploring its relation to reduction and diagonalization.

Contribution

It introduces the concept of normality for $p$-adic operators, providing criteria and an algorithm for diagonalization over local fields, linking normality, reduction, and eigenspace decomposition.

Findings

Normality of a $p$-adic operator is equivalent to diagonalizability by a unitary matrix.

Diagonalization of the reduction corresponds to a partition of unity related to the spectrum.

The paper establishes criteria for normality and diagonalization in the $p$-adic setting.

Abstract

We establish an algorithm for a criterion of the diagonalisability of a matrix over a local field by a unitary matrix. For this sake, we define the notion of normality of a $p$-adic operator, and give several criteria for the normality. We study the relation between the normality and the reduction. In the finite dimensional case, the normality of an operator is equivalent to the diagonalisability of a matrix by a unitary matrix. Therefore we also study the relation between the diagonalisability and the reduction. For example, we show that the diagonalisation of the reduction gives a partition of unity corresponding to the reduction of the spectrum, which gives a functorial lift of the eigenspace decomposition of the reduction.