Two sequences of operator monotone functions under strictly chaotic order (Q2885354)

An operator \(T\) on a Hilbert space \(H\) is called positive if \(\langle Ax, x \rangle \geq 0\) for all \(x \in H\). Let \(A,B\) be positive invertible operators on \(H\). If \(\log A > \log B\) (meaning that \(\log A - \log B\) is a positive invertible operator), then we say that \(A\) is greater than \(B\) under the strict chaotic order and denote this by \(A >> B\). Let \({\mathcal B} (H)\) denote the space of all bounded linear operators on \(H\). A function \(f: {\mathcal B} (H) \longrightarrow {\mathcal B} (H)\) is called an operator monotone function (with respect to the strict chaotic order) if \(f(A) > f(B)\) (meaning that \(f(A)-f(B)\) is a positive invertible operator) whenever \(A >> B\). The authors study two sequences of operator monotone functions using methods different from the approaches adopted earlier.