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Keywords:
commuting operators; positive selfadjoint operator; spectral representation
commuting operators; positive selfadjoint operator; spectral representation
Summary:
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.
References:
[1] Akhiezer, N. I., Glasman, I. M.: Theory of linear operators in Hilbert space. Tech. report, Vyshcha Shkola, Kharkov, 1977, English transl. Pitman (APP), 1981. MR 0486990
[2] Belaidi, B., Farissi, A. El, Latreuch, Z.: Inequalities between sum of the powers and the exponential of sum of nonnegative sequence. RGMIA Research Collection, 11 (1), Article 6, 2008.
[3] Qi, F.: Inequalities between sum of the squares and the exponential of sum of nonnegative sequence. J. Inequal. Pure Appl. Math. 8 (3) (2007), 1–5, Art. 78. MR 2345933
[4] Weidman, J.: Linear operators in Hilbert spaces. New York, Springer, 1980. MR 0566954
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