John decompositions: selecting a large part
R. Vershynin
TL;DR
The paper generalizes Bourgain–Tzafriri’s restricted invertibility to arbitrary John decompositions by proving an extraction theorem: given id = ∑ x_j ⊗ x_j and a norm-one operator T, a large subset exists on which {T x_j} behaves like an orthogonal basis with uniform lower bounds. This leads to Dvoretzky–Rogers type lemmas for contact points of X, enabling refined constructions of subspaces isomorphic to l_∞^m and yielding improved embeddings of cubes into finite-dimensional spaces. The results are connected to frames, showing that large, well-behaved subsequences exist in tight frames, and they illuminate the geometry of Banach spaces via the John decomposition and maximal volume ellipsoid framework. The work advances the local theory by linking invertibility, contact points, and finite-dimensional embeddings through a unified extraction mechanism.
Abstract
We extend the invertibility principle of J. Bourgain and L. Tzafriri to operators acting on arbitrary decompositions id = \sum x_j \otimes x_j, rather than on the coordinate one. The John's decomposition brings this result to the local theory of Banach spaces. As a consequence, we get a new lemma of Dvoretzky-Rogers type, where the contact points of the unit ball with its maximal volume ellipsoid play a crucial role. This is applied to embeddings of l_\infty^k into finite dimensional spaces.