We give a simpler proof of the Gromov–Milman theorem on concentration phenomenon on uniformly convex sphere. We also outline Rohlin’s theory of measurable partitions used in the proof. The purpose of this note is to present a localization technique for the sphere S on an example of the Gromov–Milman theorem [Gr-M] about the concentration phenomenon on uniformly convex spheres. This result was o...

Consider the linear general eigenvalue problem Ay = λBy , where A and B are both invertible and Hermitian N × N matrices. In this paper we construct a set of meromorphic functions, the Krein eigenvalues, whose zeros correspond to the real eigenvalues of the general eigenvalue problem. The Krein eigenvalues are generated by the Krein matrix, which is constructed through projections on the positi...

M. Krein proved in [KR48] that if T is a continuous operator on a normed space leaving invariant an open cone, then its adjoint T ∗ has an eigenvector. We present generalizations of this result as well as some applications to C∗-algebras, operators on l1, operators with invariant sets, contractions on Banach lattices, the Invariant Subspace Problem, and von Neumann algebras. M. Krein proved in ...

We construct rather general random ensembles that yield the optimal (isomorphic) estimate in Dvoretzky-Milman Theorem. This is first construction of non gaussian/spherical exhibit behaviour. The constructed here need not satisfy any rotation invariance and can be heavy-tailed.

The basic higher order commutator theorem is formulated for the real interpolation methods associated with the quasi-power parameters, that is, the function spaces on which Hardy inequalities are valid. This theorem unifies and extends various results given by Cwikel, Jawerth, Milman, Rochberg, and others, and incorporates some results of Kalton to the context of commutator estimates for the re...