Inequalities between the sum of powers and the exponential of sum of positive and commuting selfadjoint operators
Berrabah Bendoukha; Hafida Bendahmane
Archivum Mathematicum (2011)
- Volume: 047, Issue: 4, page 257-262
- ISSN: 0044-8753
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topBendoukha, Berrabah, and Bendahmane, Hafida. "Inequalities between the sum of powers and the exponential of sum of positive and commuting selfadjoint operators." Archivum Mathematicum 047.4 (2011): 257-262. <http://eudml.org/doc/246278>.
@article{Bendoukha2011,
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.},
author = {Bendoukha, Berrabah, Bendahmane, Hafida},
journal = {Archivum Mathematicum},
keywords = {commuting operators; positive selfadjoint operator; spectral representation; commuting operators; positive selfadjoint operator; spectral representation},
language = {eng},
number = {4},
pages = {257-262},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Inequalities between the sum of powers and the exponential of sum of positive and commuting selfadjoint operators},
url = {http://eudml.org/doc/246278},
volume = {047},
year = {2011},
}
TY - JOUR
AU - Bendoukha, Berrabah
AU - Bendahmane, Hafida
TI - Inequalities between the sum of powers and the exponential of sum of positive and commuting selfadjoint operators
JO - Archivum Mathematicum
PY - 2011
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 047
IS - 4
SP - 257
EP - 262
AB - 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.
LA - eng
KW - commuting operators; positive selfadjoint operator; spectral representation; commuting operators; positive selfadjoint operator; spectral representation
UR - http://eudml.org/doc/246278
ER -
References
top- Akhiezer, N. I., Glasman, I. M., Theory of linear operators in Hilbert space, Tech. report, Vyshcha Shkola, Kharkov, 1977, English transl. Pitman (APP), 1981. (1977) MR0486990
- 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. (2008)
- 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. (2007) MR2345933
- Weidman, J., Linear operators in Hilbert spaces, New York, Springer, 1980. (1980) MR0566954
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