Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T19:31:00.923Z Has data issue: false hasContentIssue false
  • English
  • Français

RESTRICTED INVERTIBILITY AND THE BANACH–MAZUR DISTANCE TO THE CUBE

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

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
Get access

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ball, K., Ellipsoids of maximal volume in convex bodies. Geom. Dedicata 41 (2) (1992), 241–250.Google Scholar
Batson, J. D., Spielman, D. A. and Srivastava, N., Twice-Ramanujan sparsifiers. In STOC’09 – Proceedings of the 2009 ACM International Symposium on Theory of Computing, ACM (New York, 2009), 255–262.Google Scholar
Bourgain, J. and Szarek, S. J., The Banach–Mazur distance to the cube and the Dvoretzky–Rogers factorization. Israel J. Math. 62 (2) (1988), 169–180.CrossRefGoogle Scholar
Bourgain, J. and Tzafriri, L., Invertibility of large submatrices with applications to the geometry of banach spaces and harmonic analysis. Israel J. Math. 57 (1987), 137–224.CrossRefGoogle Scholar
Diestel, J., Jarchow, H. and Tonge, A., Absolutely Summing Operators (Cambridge Studies in Advanced Mathematics 43) Cambridge University Press (Cambridge, 1995).CrossRefGoogle Scholar
Dvoretzky, A. and Rogers, C. A., Absolute and unconditional convergence in normed linear spaces. Proc. Natl. Acad. Sci. USA 36 (1950), 192–197.CrossRefGoogle ScholarPubMed
Giannopoulos, A. A., A note on the Banach–Mazur distance to the cube. In Geometric Aspects of Functional Analysis (Israel, 1992–1994) (Operator Theory: Advances and Applications 77) Birkhäuser (Basel, 1995), 67–73.Google Scholar
Giannopoulos, A. A., A proportional Dvoretzky–Rogers factorization result. Proc. Amer. Math. Soc. 124 (1) (1996), 233–241.CrossRefGoogle Scholar
Gluskin, E. D., The diameter of the Minkowski compactum is roughly equal to $n$. Funktsional. Anal. i Prilozhen. 15 (1) (1981), 72–73.Google Scholar
John, F., Extremum problems with inequalities as subsidiary conditions. In Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc (New York, NY, 1948), 187–204.Google Scholar
Kashin, B. and Tzafriri, L., Some remarks on the restriction of operators to coordinate subspaces. Preprint, 1993.Google Scholar
Ledoux, M. and Talagrand, M., Probability in Banach spaces, Springer (Berlin, 1991).Google Scholar
Litvak, A. E. and Tomczak-Jaegermann, N., Random aspects of high-dimensional convex bodies. In Geometric Aspects of Functional Analysis (Lecture Notes in Mathematics 1745) Springer (Berlin, 2000), 169–190.Google Scholar
Lunin, A. A., On operator norms of submatrices. Mat. Zametki 45 (3) (1989), 94–100, 128.Google Scholar
Spielman, D. A. and Srivastava, N., An elementary proof of the restricted invertibility theorem. Israel J. Math. 190 (2012), 83–91.CrossRefGoogle Scholar
Srivastava, N., Spectral sparsification and restricted invertibility. PhD Thesis, Yale University, March 2010.Google Scholar
Szarek, S. J., Spaces with large distance to ${ l}_{\infty }^{n} $ and random matrices. Amer. J. Math. 112 (6) (1990), 899–942.CrossRefGoogle Scholar
Szarek, S. J. and Talagrand, M., An isomorphic version of the Sauer-Shelah lemma and the Banach–Mazur distance to the cube. In Geometric Aspects of Functional Analysis (1987–88) (Lecture Notes in Mathematics 1376) Springer (Berlin, 1989), 105–112.Google Scholar
Taschuk, S., The Banach–Mazur distance to the cube in low dimensions. Discrete Comput. Geom. 46 (1) (2011), 175–183.Google Scholar
Tropp, J. A., Column subset selection, matrix factorization, and eigenvalue optimization. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM (Philadelphia, PA, 2009), 978–986.Google Scholar
Vershynin, R., John’s decompositions: selecting a large part. Israel J. Math. 122 (2001), 253–277.CrossRefGoogle Scholar