Comptes Rendus
no. 10
Functional analysis
Positive block matrices and numerical ranges
[Matrices par blocs positives et images numériques]

Présenté par : the Editorial Board

Jean-Christophe Bourin1 ; Antoine Mhanna2
1 Laboratoire de mathématiques de Besançon, Université Bourgogne Franche-Comté, CNRS UMR 6623, 16, route de Gray, 25030 Besançon, France
2 Kfardebian, Lebanon
Comptes Rendus. Mathématique, Volume 355 (2017) no. 10, pp. 1077-1081.

Toute matrice positive partitionnée en quatre blocs de même taille satisfait l'inégalité en norme unitairement invariante ‖M‖≤‖M1,1+M2,2+ωI‖, où ω est la largeur de l'image numérique de M1,2.

Any positive matrix M partitioned in four n-by-n blocks satisfies the unitarily invariant norm inequality ‖M‖≤‖M1,1+M2,2+ωI‖, where ω is the width of the numerical range of M1,2. Some related inequalities and a reverse Lidskii majorization are given.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.10.006
Jean-Christophe Bourin 1 ; Antoine Mhanna 2

1 Laboratoire de mathématiques de Besançon, Université Bourgogne Franche-Comté, CNRS UMR 6623, 16, route de Gray, 25030 Besançon, France
2 Kfardebian, Lebanon 
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Jean-Christophe Bourin; Antoine Mhanna. Positive block matrices and numerical ranges. Comptes Rendus. Mathématique, Volume 355 (2017) no. 10, pp. 1077-1081. doi : 10.1016/j.crma.2017.10.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.10.006/

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[3] J.-C. Bourin; E.-Y. Lee; M. Lin On a decomposition lemma for positive semi-definite block-matrices, Linear Algebra Appl., Volume 437 (2012), pp. 1906-1912

[4] H. Du; C.-K. Li; K.-Z. Wang; Y. Wang; N. Zuo Numerical ranges of the product of operators, Oper. Matrices, Volume 11 (2017) no. 1, pp. 171-180

[5] T. Hiroshima Majorization criterion for distillability of a bipartite quantum state, Phys. Rev. Lett., Volume 91 (2003) no. 5

[6] F. Kittaneh Norm inequalities for certain operator sums, J. Funct. Anal., Volume 143 (1997), pp. 337-348

[7] F. Kittaneh Norm inequalities for commutators of positive operators and applications, Math. Z., Volume 258 (2008), pp. 845-849

[8] H. Klaja The numerical range and the spectrum of a product of two orthogonal projections, J. Math. Anal. Appl., Volume 411 (2014), pp. 177-195

[9] M. Lin; H. Wolwowicz Hiroshima's theorem and matrix norm inequalities, Acta Sci. Math. (Szeged), Volume 81 (2015) no. 1–2, pp. 45-53

[10] A. Mhanna On symmetric norm inequalities and positive definite block-matrices, Math. Inequal. Appl. (2017) (in press)

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