Nicole Tomczak-Jaegermann

Projections onto Hilbertian subspaces of Banach spaces

In this paper we obtain new estimates for the relative projection constants of subspaces of a Banach spaceY in terms of geometrical properties ofY. Our method gives thatK-convex spaces are locally π-Euclidean. We also get a version of Maurey’s extension theorem for spaces of typep<2.

Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles

Let K be an isotropic convex body in Rn. Given e > 0, how many independent points Xi uniformly distributed on K are neededfor the empirical covariance matrix to approximate the identity up to e with overwhelming probability? Our paper answers this question from [12]. More precisely, let X ∈ Rn be a centered random vector with a log-concave distribution and with the identity as covariance matrix. An example of such a vector X is a random point in an isotropic convex body. We show that for any e > 0, there existsC(e) > 0, suﰁch that if N ∼ C(e)n and (Xi)i≤N are i.i.d. copies of ﱞﱞ1 N ﱞﱞ X, then ﱞN i=1 Xi ⊗ Xi − Idﱞ ≤ e, with probability larger than 1 − exp(−c√n).

Banach spaces without local unconditional structure

For a large class of Banach spaces, a general construction of subspaces without local unconditional structure is presented. As an application it is shown that every Banach space of finite cotype contains eitherl2 or a subspace without unconditional basis, which admits a Schauder basis. Some other interesting applications and corollaries follow.

Asymptotic Infinite-Dimensional Theory of Banach Spaces

In this paper we study structural properties of infinite dimensional Banach spaces. The classical understanding of such properties was developed in the 50s and 60s; goals of the theory had direct roots in and were natural expansion of problems from the times of Banach. Most of surveys and books of that period directly or indirectly discussed such problems as the existence of unconditional basic sequences, the c0-l 1-reflexive subspace problem and others. However, it has been realized recently that such a nice and elegant structural theory does not exist. Recent examples (or counter-examples to classical problems) due to Gowers and Maurey [GM] and Gowers [G.2], [G.3] showed much more diversity in the structure of infinite dimensional subspaces of Banach spaces than was expected.

Projection constants of symmetric spaces and variants of Khintchine's inequality

Abstract The projection constants of the lpn-spaces for 1 ≦ p ≦ 2 satisfy with in the real case and in the complex case. Further, there is c < 1 such that the projection constant of any n-dimensional space Xn with 1-symmetric basis can be estimated by . The proofs of the results are based on averaging techniques over permutations and a variant of Khintchines inequality which states that

Banach spaces of typep have arbitrarily distortable subspaces

It is shown that if a Banach space has bounded distortions then it contains an unconditional basic sequence. It follows that Banach spaces of typep > 1 contain arbitrarily distortable subspaces. Furthermore, hereditarily indecomposable Banach spaces are themselves arbitrarily distortable.

Uniform convexity of unitary ideals

ItE is a symmetric Banach sequence which isq-concave with the constant equal to 1 (where 2≦q<∞), thenSE isq-PL-convex. IfE isq-concave andp-convex with the constants equal to 1 (where 1<p≦2≦q<∞), thenSE is uniformly convex with modulus of convexity of power typeq and uniformly smooth with modulus of smoothness of power typep.

Euclidean embeddings in spaces of finite volume ratio via random matrices

Abstract Let (ℝ n , || ⋅ ||) be the space ℝ N equipped with a norm || ⋅ || whose unit ball has a bounded volume ratio with respect to the Euclidean unit ball. Let Γ be any random N  × n matrix with N  > n , whose entries are independent random variables satisfying some moment assumptions. We show that with high probability Γ is a good isomorphism from the n -dimensional Euclidean space (ℝ N , | ⋅ |) onto its image in (ℝ N , || ⋅ ||) , i.e. there exist α, β > 0 such that for all x ∈ ℝ N , . This solves a conjecture of Schechtman on random embeddings of ℓ 2 n into ℓ 1 N .

A Short Proof of Paouris' Inequality

We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm

Tail estimates for norms of sums of log-concave random vectors

|X|