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Approximate John’s Decompositions

  • M. Rudelson
Conference paper
Part of the Operator Theory Advances and Applications book series (OT, volume 77)

Abstract

Let X be an n-dimensional Banach space such that the ellipsoid of minimal volume containing its unit ball B X is the standard Euclidean ball 2 n . By the well-known theorem of F. John [J], there exists a decomposition of the identity operator of X
$$ id = \sum\limits_{i = 1}^k {{c_i}{x_i} \otimes {x_i}} $$
(1)
where c i > 0, x i are contact points (∥x i x = ∥x i 2= 1) and kn·(n+l)/2 in the real and kn 2 in the complex case. This decomposition is often unique. Moreover, Pelczynski and Tomczak-Jaegermann proved that for every given k, nkN, there exists an n-dimensional Banaeh space X for which (1) holds with this k and this representation is unique if x i are contact points of John’s ellipsoid of X [P-T-J]. However, if we want only to approximate the identity operator, the length k of this decomposition can always be reduced to a number close to n.

Keywords

Contact Point Unit Ball Identity Operator Bernoulli Variable Finite Rank Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [J]
    John, F., Extremum problems with inequalities as subsidiary conditions, in Courant Anniversary Volume, Interscienee, New York, 1948, 187–204.Google Scholar
  2. [L-T]
    Ledoux, M., Talagrand, M., Probability in Banach spaces, Ergeb. Math. Grenzgeb., 3 Folge, vol. 23, Springer, Berlin, 1991.Google Scholar
  3. [Pa-T-J]
    Pajor, A., Tomczak-Jaegermann, N., Subspaces of small codimension of finite dimensional Banach spaces, Proc. Amer. Math. Soc. 97 (1986), 637–642.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [P-T-J]
    Pelczynski, A., Tomczak-Jaegermann, N., On the length of faithful nuclear representations of finite rank operators, Mathematika 35 (1988), no. 1, 126–143.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 1995

Authors and Affiliations

  • M. Rudelson
    • 1
  1. 1.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael

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