Noncommutative Khintchine inequalities in interpolation spaces of LpL_p-spaces

We prove noncommutative Khintchine inequalities for all interpolation spaces between $L_p$ and $L_2$ with $p<2$. In particular, it follows that Khintchine inequalities hold in $L_{1,\infty}$. Using a similar method, we find a new deterministic equivalent for the $RC$-norm in all interpolation spaces between $L_p$-spaces which unifies the cases $p > 2$ and $p < 2$. It produces a new proof of Khintchine inequalities for $p<1$ for free variables. To complete the picture, we exhibit counter-examples which show that neither of the usual closed formulas for Khintchine inequalities can work in $L_{2,\infty}$. We also give an application to martingale inequalities.Comment: 33 pages, published versio

Spectral gap properties of the unitary groups: around Rider's results on non-commutative Sidon sets

We present a proof of Rider's unpublished result that the union of two Sidon sets in the dual of a non-commutative compact group is Sidon, and that randomly Sidon sets are Sidon. Most likely this proof is essentially the one announced by Rider and communicated in a letter to the author around 1979 (lost by him since then). The key fact is a spectral gap property with respect to certain representations of the unitary groups $U(n)$ that holds uniformly over $n$. The proof crucially uses Weyl's character formulae. We survey the results that we obtained 30 years ago using Rider's unpublished results. Using a recent different approach valid for certain orthonormal systems of matrix valued functions, we give a new proof of the spectral gap property that is required to show that the union of two Sidon sets is Sidon. The latter proof yields a rather good quantitative estimate. Several related results are discussed with possible applications to random matrix theory.Comment: v2: minor corrections, v3 more minor corrections v4) minor corrections, last section removed to be included in another paper in preparation with E. Breuillard v5) more minor corrections + two references added. The paper will appear in a volume dedicated to the memory of V. P. Havi

Mini-Workshop: Modern Applications of s-numbers and Operator Ideals

The main aim of this mini-workshop was to present and discuss some modern applications of the functional-analytic concepts of $s$-numbers and operator ideals in areas like Numerical Analysis, Theory of Function Spaces, Signal Processing, Approximation Theory, Probability on Banach Spaces, and Statistical Learning Theory

Lorentz-Shimogaki-Arazy-Cwikel Theorem Revisited

We present a new approach to Lorentz-Shimogaki and Arazy-Cwikel Theorems which covers all range of $p,q\in (0,\infty]$ for function spaces and sequence spaces. As a byproduct, we solve a conjecture of Levitina and the last two authors.Comment: 27 page

Families of completely positive maps associated with monotone metrics

An operator convex function on (0,\infty) which satisfies the symmetry condition k(1/x) = x k(x) can be used to define a type of non-commutative multiplication by a positive definite matrix (or its inverse) using the primitive concepts of left and right multiplication and the functional calculus. The operators for the inverse can be used to define quadratic forms associated with Riemannian metrics which contract under the action of completely positive trace-preserving maps. We study the question of when these operators define maps which are also completely positive (CP). Although A --> D^{-1/2} A D^{-1/2} is the only case for which both the map and its inverse are CP, there are several well-known one parameter families for which either the map or its inverse is CP. We present a complete analysis of the behavior of these families, as well as the behavior of lines connecting an extreme point with the smallest one and some results for geometric bridges between these points. Our primary tool is an order relation based on the concept of positive definite functions. Although some results can be obtained from known properties, we also prove new results based on the positivity of the Fourier transforms of certain functions. Concrete computations of certain Fourier transforms not only yield new examples of positive definite functions, but also examples in the much stronger class of infinitely divisible functions.Comment: Final version to appear in Lin Alg. Appl. Links to preprints adde

Square functions and radonifying operators

Peer reviewe

Analysis in Banach Spaces Volume II : Probabilistic Methods and Operator Theory Preface

Peer reviewe

Positive Definite Matrices: Compression, Decomposition, Eigensolver, and Concentration

For many decades, the study of positive-definite (PD) matrices has been one of the most popular subjects among a wide range of scientific researches. A huge mass of successful models on PD matrices has been proposed and developed in the fields of mathematics, physics, biology, etc., leading to a celebrated richness of theories and algorithms. In this thesis, we draw our attention to a general class of PD matrices that can be decomposed as the sum of a sequence of positive-semidefinite matrices. For this class of PD matrices, we will develop theories and algorithms on operator compression, multilevel decomposition, eigenpair computation, and spectrum concentration. We divide these contents into three main parts. In the first part, we propose an adaptive fast solver for the preceding class of PD matrices which includes the well-known graph Laplacians. We achieve this by establishing an adaptive operator compression scheme and a multiresolution matrix factorization algorithm which have nearly optimal performance on both complexity and well-posedness. To develop our methods, we introduce a novel notion of energy decomposition for PD matrices and two important local measurement quantities, which provide theoretical guarantee and computational guidance for the construction of an appropriate partition and a nested adaptive basis. In the second part, we propose a new iterative method to hierarchically compute a relatively large number of leftmost eigenpairs of a sparse PD matrix under the multiresolution matrix compression framework. We exploit the well-conditioned property of every decomposition components by integrating the multiresolution framework into the Implicitly Restarted Lanczos method. We achieve this combination by proposing an extension-refinement iterative scheme, in which the intrinsic idea is to decompose the target spectrum into several segments such that the corresponding eigenproblem in each segment is well-conditioned. In the third part, we derive concentration inequalities on partial sums of eigenvalues of random PD matrices by introducing the notion of k-trace. For this purpose, we establish a generalized Lieb's concavity theorem, which extends the original Lieb's concavity theorem from the normal trace to k-traces. Our argument employs a variety of matrix techniques and concepts, including exterior algebra, mixed discriminant, and operator interpolation.</p