### Adjointations of Operator Inequalities and Characterizations of Operator Monotonicity via Operator Means

Pattrawut Chansangiam

#### Abstract

We propose adjointations between operator orderings, which convert any operator inequalities/identities associated with certain binary operations to new ones. Then we prove that a continuous function $$f:(0,\infty) \to (0,\infty)$$ is operator monotone increasing if and only if $$f(A \: !_t \: B) \leq f(A) \: !_t \: f(B)$$ 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.

#### Keywords

Operator mean; Operator monotone function; Operator inequality

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#### References

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DOI: http://dx.doi.org/10.26713%2Fcma.v7i2.372

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