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

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.