Upper and lower bounds for an integral transform of positive operators in Hilbert spaces with applications

Abstract

For a continuous and positive function \(w(\lambda)\), \(\lambda>0\) and a positive measure \(\mu\) on \((0,\infty )\) we consider the following integral transform
\[
\mathcal{D}( w,\mu ) ( T) :=\int_{0}^{\infty }w(\lambda) (\lambda +T)^{-1} d\mu ( \lambda ) ,
\]
where the integral is assumed to exist for any positive operator \(T\) on a complex Hilbert space \(H\). In this paper we obtain several upper and lower bounds for the difference \(\mathcal{D}( w,\mu ) ( A) -\mathcal{D}( w,\mu ) ( B)\) under certain assumptions for the operators \(A\) and \(B\). Some natural applications for operator monotone and operator convex functions are also given.