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arXiv · 1506.06922

Characterizations of Operator Monotonicity via Operator Means and Applications to Operator Inequalities

Pattrawut Chansangiam
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Abstract

We prove that a continuous function $f:(0,\infty) \to (0,\infty)$ is operator monotone increasing if and only if $f(A \: !_t \: B) \leqs f(A) \: !_t \: f(B)$ for any positive operators $A,B$ and scalar $t \in [0,1]$. Here, $!_t$ denotes the $t$-weighted harmonic mean. As a counterpart, $f$ is operator monotone decreasing if and only if the reverse of preceding inequality holds. Moreover, we obtain many characterizations of operator-monotone increasingness/decreasingness in terms of operator means. These characterizations lead to many operator inequalities involving means.

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Pattrawut Chansangiam. 2015-06-23. Characterizations of Operator Monotonicity via Operator Means and Applications to Operator Inequalities. https://arxiv.org/abs/1506.06922

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