677 research outputs found
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
, whose invariant density 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 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 from E(\M) into a Hilbert space
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
denotes the 2-concavification of and is a universal constant.
As a consequence we obtain the noncommutative Khintchine inequalities for
E(\M) when is either 2-concave or 2-convex and -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
- …