An extension of the Lowner-Heinz inequality

TL;DR

This paper extends the L"owner--Heinz inequality to a broader class of operators, providing new bounds for operator differences and their logarithms, with implications for operator theory.

Contribution

It introduces a novel inequality for operator powers and logarithms, extending the classical L"owner--Heinz inequality to operators with a specific ordering.

Findings

Established a new inequality for A^r - B^r with explicit bounds.

Proved a related inequality for the difference of logarithms of operators.

Demonstrated the positivity of these operator differences under certain conditions.

Abstract

We extend the celebrated L\"owner--Heinz inequality by showing that if $A, B$ are Hilbert space operators such that $A > B \geq 0$, then A^r - B^r \geq ||A||^r-(||A||- \frac{1}{||(A-B)^{-1}||})^r > 0 for each $0 < r \leq 1$. As an application we prove that \log A - \log B \geq \log||A||- \log(||A||-\frac{1}{||(A-B)^{-1}||})>0.