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arXiv · 1811.00475

Operator Ky Fan type inequalities

S. Habibzadeh·J. Rooin·M. S. Moslehian
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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 $f:[0,\infty)\rightarrow[0,\infty)$ is an operator monotone function with $f (1) = 1$, $f'(1)=\mu$, and associated mean $\sigma$, then for all operators $A$ and $B$ on a complex Hilbert space $\mathscr{H}$ such that $0<A,B\leq\frac{1}{2}I$, we have \begin{equation*} A'\nabla_\mu B'-A'\sigma B'\leq A\nabla_\mu B-A\sigma B, \end{equation*} where $I$ is the identity operator on $\mathscr{H}$, $A':=I-A$, $B':=I-B$, and $\nabla_\mu$ is the $\mu$-weighted arithmetic mean.

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S. Habibzadeh, J. Rooin, M. S. Moslehian. 2018-11-01. Operator Ky Fan type inequalities. https://doi.org/10.1016/j.laa.2018.07.009

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