Geometric Aspects of Functional Analysis pp 245-249 | Cite as

# Approximate John’s Decompositions

Conference paper

## Abstract

Let where

*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*ℓ*_{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)

*c*_{ i }> 0,*x*_{ i }are contact points (∥*x*_{ i }∥*x*= ∥*x*_{ i }∥_{2}= 1) and*k*≤*n*·(*n*+l)/2 in the real and*k*≤*n*^{2}in the complex case. This decomposition is often unique. Moreover, Pelczynski and Tomczak-Jaegermann proved that for every given*k*,*n*≤*k*≤*N*, 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

- [J]John, F.,
*Extremum problems with inequalities as subsidiary conditions*, in Courant Anniversary Volume, Interscienee, New York, 1948, 187–204.Google Scholar - [L-T]Ledoux, M., Talagrand, M.,
*Probability in Banach spaces*, Ergeb. Math. Grenzgeb., 3 Folge, vol. 23, Springer, Berlin, 1991.Google Scholar - [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 - [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