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

#### Abstract

\[

\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.

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DOI: http://dx.doi.org/10.17951/a.2022.76.1.1-15

Data publikacji: 2022-10-05 20:39:29

Data złożenia artykułu: 2022-10-04 18:39:31

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