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

Abstract

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Let ( ) be the set of all bounded linear operators acting in Hilbert space and + ( ) the set of all positive selfadjoint elements of ( ) . The aim of this paper is to prove that for every finite sequence ( A i ) i = 1 n of selfadjoint, commuting elements of + ( ) and every natural number p 1 , the inequality e p p p i = 1 n A i p exp i = 1 n A i , holds.

How to cite

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Bendoukha, 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

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  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. (1977) MR0486990
  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. (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. (2007) MR2345933
  4. Weidman, J., Linear operators in Hilbert spaces, New York, Springer, 1980. (1980) MR0566954

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