Effective Construction of a Positive Operator which does not admit Triangular Factorization

TL;DR

This paper presents a concrete example of a positive operator that cannot be factorized, providing explicit counterexamples to well-known open problems in operator theory.

Contribution

The authors construct a specific non-factorable positive operator, replacing abstract existence theorems with explicit examples for key problems.

Findings

Provided a concrete non-factorable positive operator

Challenged assumptions in the Ringrose, Kadison, and Singer problems

Replaced existence theorems with explicit counterexamples

Abstract

We have constructed a concrete example of a non-factorable positive operator. As a result, for the well-known problems (Ringrose, Kadison and Singer problems) we replace existence theorems by concrete examples.