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

Silvestru Sever Dragomir


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.


Operator monotone functions; operator convex functions; operator inequalities; logarithmic operator inequalities

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DOI: http://dx.doi.org/10.17951/a.2022.76.1.1-15
Date of publication: 2022-10-05 20:39:29
Date of submission: 2022-10-04 18:39:31


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