Krein-\v{S}mul'jan Theorem Revisited

TL;DR

This paper generalizes the Krein-mul'jan theorem to multiple operators, providing conditions to identify when a linear combination of bounded selfadjoint operators is positive semidefinite.

Contribution

It extends the classical Krein-mul'jan theorem to multiple operators, offering new criteria for positive semidefiniteness of their linear combinations.

Findings

Derived sufficient conditions for positive semidefiniteness involving multiple operators

Generalized the Krein-mul'jan theorem to a broader setting

Provided theoretical framework for operator positivity analysis

Abstract

We present a generalization of Krein-\v{S}mul'jan theorem which involves several operators. Given bounded selfadjoint operators $A,B_1,\ldots,B_m$ acting on a Hilbert space $\mathcal{H}$, we provide sufficient conditions to determine whether there are $\lambda_1,\ldots,\lambda_m\in \mathbb{R}$ such that $A + \sum_{i=1}^m \lambda_i B_i$ is a positive semidefinite operator.