Inequalities between the sum of powers and the exponential of sum of positive and commuting selfadjoint operators

Berrabah Bendoukha and Hafida Bendahmane

Address: Laboratoire de Mathématiques pures et appliquées, Abdelhmid Ibn Badis Mostaganem University, B.O. 227, Mostaganem (27000), Algeria


Abstract: Let ${\mathcal{B}}({\mathcal{H}})$ be the set of all bounded linear operators acting in Hilbert space ${\mathcal{H}}$ and ${\mathcal{B}}^{+}({\mathcal{H}})$ the set of all positive selfadjoint elements of ${\mathcal{B}}({\mathcal{H}})$. The aim of this paper is to prove that for every finite sequence $(A_{i})_{i=1}^{n}$ of selfadjoint, commuting elements of ${\mathcal{B}}^{+}({\mathcal{H}})$ and every natural number $p\ge 1$, the inequality \[ \frac{e^{p}}{p^{p}}\Big (\sum _{i=1}^{n}A_{i}^{p}\Big )\le \exp \Big (\sum _{i=1}^{n}A_{i}\Big )\,, \] holds.

AMSclassification: primary 47B60; secondary 47A30.

Keywords: commuting operators, positive selfadjoint operator, spectral representation.