Ornstein-Uhlenbeck semi-groups on stratified groups

We consider, in the setting of stratified groups G, two analogues of the Ornstein-Uhlenbeck semi-group, namely Markovian diffusion semi-groups acting on $L^q(pd g)$, whose invariant density $p$ is a heat kernel at time 1 on G

The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators

We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let \M be a von Neumann algebra equipped with a normal faithful semifinite trace \t, and let $E$ be an r.i. space on (0, \8). Let E(\M) be the associated symmetric space of measurable operators. Then to any bounded linear map $T$ from E(\M) into a Hilbert space $\mathcal H$ corresponds a positive norm one functional f\in E_{(2)}(\M)^* such that \forall x\in E(\M)\quad \|T(x)\|^2\le K^2 \|T\|^2 f(x^*x+xx^*), where $E_{(2)}$ denotes the 2-concavification of $E$ and $K$ is a universal constant. As a consequence we obtain the noncommutative Khintchine inequalities for E(\M) when $E$ is either 2-concave or 2-convex and $q$-concave for some q<\8. We apply these results to the study of Schur multipliers from a 2-convex unitary ideal into a 2-concave one.Comment: 14 pages. To appear in J. Funct. Ana

Matrix probing and its conditioning

When a matrix A with n columns is known to be well approximated by a linear combination of basis matrices B_1,..., B_p, we can apply A to a random vector and solve a linear system to recover this linear combination. The same technique can be used to recover an approximation to A^-1. A basic question is whether this linear system is invertible and well-conditioned. In this paper, we show that if the Gram matrix of the B_j's is sufficiently well-conditioned and each B_j has a high numerical rank, then n {proportional} p log^2 n will ensure that the linear system is well-conditioned with high probability. Our main application is probing linear operators with smooth pseudodifferential symbols such as the wave equation Hessian in seismic imaging. We demonstrate numerically that matrix probing can also produce good preconditioners for inverting elliptic operators in variable media