Operator Ky Fan type inequalities

Abstract

In this paper, we extend some significant Ky Fan type inequalities in a large setting to operators on Hilbert spaces and derive their equality conditions. Among other things, we prove that if $f:[0,\infty )\to [0,\infty )$ is an operator monotone function with $f(1)=1$, $f^{\prime }(1)=\mu$, and associated mean $\sigma$, then for all operators $A$ and $B$ on a complex Hilbert space $\mathcal{H}$ such that $0<A,B\le \frac{1}{2}I$, we have \begin{equation*} A'\nabla_μB'-A'σB'\leq A\nabla_μB-AσB, \end{equation*} where $I$ is the identity operator on $\mathcal{H}$, $A^{\prime }:=I-A$, $B^{\prime }:=I-B$, and ${\nabla }_{\mu }$ is the $\mu$-weighted arithmetic mean.

14 pages (Linear Algebra Appl.)

References in corpus (3)

• Integral Representations and Decompositions of Operator Monotone Functions on the Nonnegative Reals
• Alzer Inequality for Hilbert Spaces Operators
• Operator revision of a Ky Fan type inequality