Effective construction of a class of positive operators in Hilbert space, which do not admit triangular factorization

TL;DR

This paper constructs a specific class of positive operators in Hilbert space that cannot be factorized triangularly, providing concrete examples to address longstanding theoretical questions.

Contribution

It introduces a new class of positive operators in Hilbert space that lack triangular factorization, replacing abstract existence theorems with explicit examples.

Findings

Constructed a class of non-factorable positive operators.

Provided concrete examples for classical problems.

Challenged assumptions about operator factorization.

Abstract

A class of non-factorable positive operators is constructed. As a result, pure existence theorems in the well-known problems by Ringrose, Kadison and Singer are substituted by concrete examples.