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

Bendoukha, Berrabah ; Bendahmane, Hafida

Archivum mathematicum, Tome 47 (2011) no. 4, pp. 257-262.

Résumé

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.

Détail

Classification : 47A30, 47B60
Keywords: commuting operators; positive selfadjoint operator; spectral representation