Convex $\operatorname{SO}(N)\times \operatorname{SO}(n)$-invariant functions and refinements of von Neumann’s inequality

Dacorogna, Bernard; Maréchal, Pierre

doi:10.5802/afst.1139

Convex

SO (N) \times SO (n)

-invariant functions and refinements of von Neumann’s inequality

Dacorogna, Bernard; Maréchal, Pierre

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 16 (2007) no. 1, p. 71-89

Abstract
Résumé

A function $f$ on $M_{N \times n} (ℝ)$ which is $SO (N) \times SO (n)$ -invariant is convex if and only if its restriction to the subspace of diagonal matrices is convex. This results from Von Neumann type inequalities and appeals, in the case where $N = n$ , to the notion of signed singular value.

Une fonction $f$ sur $M_{N \times n} (ℝ)$ qui est $SO (N) \times SO (n)$ -invariante est convexe si et seulement si sa restriction au sous-espace des matrices diagonales est convexe. Ceci résulte de variantes de l’inégalité de Von Neumann et fait appel, dans le cas où $N = n$ , à la notion de valeur singulière signée.

MR 2325592 | Zbl pre05247239

DOI : https://doi.org/10.5802/afst.1139

@article{AFST_2007_6_16_1_71_0,
     author = {Dacorogna, Bernard and Mar\'echal, Pierre},
     title = {Convex $\operatorname{SO}(N)\times \operatorname{SO}(n)$-invariant functions and refinements of von Neumann's inequality},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 16},
     number = {1},
     year = {2007},
     pages = {71-89},
     doi = {10.5802/afst.1139},
     mrnumber = {2325592},
     zbl = {pre05247239},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2007_6_16_1_71_0}
}

Dacorogna, Bernard; Maréchal, Pierre. Convex $\operatorname{SO}(N)\times \operatorname{SO}(n)$-invariant functions and refinements of von Neumann’s inequality. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 16 (2007) no. 1, pp. 71-89. doi : 10.5802/afst.1139. http://www.numdam.org/item/AFST_2007_6_16_1_71_0/

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