Adjointations of Operator Inequalities and Characterizations of Operator Monotonicity via Operator Means - Chansangiam, Pattrawut (Communications in Mathematics and Applications, 2016)
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 \(f:(0,∞) → (0,∞)\) is operator monotone increasing if and only if \(f(A \: !_t \: B) ≤ f(A) \: !_t \: f(B)\) for any positive operators \(A,B\) and scalar \(t [0,1]\). 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.