Representation theorems for operators on free Banach spaces of type countable

TL;DR

This paper develops a spectral measure-based representation theorem for operators on free Banach spaces of countable type, linking algebraic structures to continuous functions and matrix representations.

Contribution

It introduces a spectral measure framework and matrix representation for operators on free Banach spaces, extending Gelfand theory to non-archimedean settings.

Findings

Operators are isometrically isomorphic to continuous functions on a compact set.

Spectral measures enable integral-based operator representations.

Matrix representations use scalar measure integrals.

Abstract

This work will be centered in commutative Banach subalgebras of the algebra of bounded linear operators defined on a Free Banach spaces of countable type. The main goal of this work wil be to formulate a representation theorem for these operators through integrals defined by spectral measures type. In order to get this objective, we will show that, under special conditions, each one of these algebras is isometrically isomorphic to some space of continuous functions defined over a compact set. Then, we will identify such compact developing the Gelfand space theory in the non-archimedean setting. This fact will allow us to define a measure which is known as spectral measure. As a second goal, we will formulate a matrix representation theorem for this class of operators whose entries of these matrices will be integrals coming from scalar measures.