A negative answer to the question of the linearity of Tate's Trace for the sum of two endomorphisms

TL;DR

This paper provides a counter-example demonstrating that Tate's trace is not linear when summing two finite potent operators on an infinite-dimensional vector space, addressing a long-standing open problem from 1968.

Contribution

It offers the first known counter-example to the linearity of Tate's trace for the sum of two operators, resolving a problem posed by J. Tate in 1968.

Findings

Counter-example shows non-linearity of Tate's trace

Addresses a 1968 open problem by Tate

Clarifies limitations of Tate's trace in infinite dimensions

Abstract

The aim of this note is to solve a problem proposed by J. Tate in 1968 by offering a counter-example of the linearity of the trace for the sum of two finite potent operators on an infinite-dimensional vector space.