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RESTRICTED INVERTIBILITY AND THE BANACH–MAZUR DISTANCE TO THE CUBE

Part of: Basic linear algebra Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  02 September 2013

Pierre Youssef
Pierre Youssef*
Affiliation:
Laboratoire d’Analyse et Mathématiques Appliquées, Université Paris-Est Marne-La-Vallée, 5, boulevard Descartes, Champs sur Marne, 77454 Marne-la-Vallée Cedex 2,France email pierre.youssef@univ-mlv.fr
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Abstract

We prove a normalized version of the restricted invertibility principle obtained by Spielman and Srivastava in [An elementary proof of the restricted invertibility theorem. Israel J. Math. 190 (2012), 83–91]. Applying this result, we get a new proof of the proportional Dvoretzky–Rogers factorization theorem recovering the best current estimate in the symmetric setting while we improve the best known result in the non-symmetric case. As a consequence, we slightly improve the estimate for the Banach–Mazur distance to the cube: the distance of every $n$-dimensional normed space from ${ \ell }_{\infty }^{n} $ is at most $\mathop{(2n)}\nolimits ^{5/ 6} $. Finally, using tools from the work of Batson et al in [Twice-Ramanujan sparsifiers. In STOC’09 – Proceedings of the 2009 ACM International Symposium on Theory of Computing, ACM (New York, 2009), 255–262], we give a new proof for a theorem of Kashin and Tzafriri [Some remarks on the restriction of operators to coordinate subspaces. Preprint, 1993] on the norm of restricted matrices.

MSC classification

Secondary: 15A09: Matrix inversion, generalized inverses 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Mathematika , Volume 60 , Issue 1 , January 2014 , pp. 201 - 218
Copyright
Copyright © University College London 2013 

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