Geodesic
Inequalities between the sum of powers and the exponential of sum of positive and commuting selfadjoint operators
Archivum mathematicum, Tome 47 (2011) no. 4, pp. 257-262 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 47A30, 47B60
Keywords: commuting operators; positive selfadjoint operator; spectral representation 
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Bendoukha, Berrabah; Bendahmane, Hafida. Inequalities between the sum of powers and the exponential of sum of positive and commuting selfadjoint operators. Archivum mathematicum, Tome 47 (2011) no. 4, pp. 257-262. http://geodesic.mathdoc.fr/item/ARM_2011_47_4_a2/

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