Operator means and positivity of block operators

Abstract

The aim of this paper is to find some sufficient conditions for positivity of block matrices of positive operators. It is shown that for positive operators A, B, C and for every non-negative operator monotone function f on $$ (0,\infty )$$ , the block matrix $$\begin{aligned} \left( \begin{array}{cc} f(A) &{} f(C) \\ f(C) &{} f(B) \\ \end{array} \right) \end{aligned}$$ is positive if and only if $$C \le A!B$$ . In particular, if $$C \le A!B$$ then $$\begin{aligned} \left( \begin{array}{cc} A &{} C \\ C &{} B \\ \end{array} \right) , \end{aligned}$$ is positive.