Concentration of mass on the Schatten classes Academic Article uri icon

abstract

  • Let 1 ≤ p ≤ ∞ and over(B (S pn ), ̃) be the unit ball of the Schatten trace class of matrices on C n or on R n , normalized to have Lebesgue measure equal to one. We prove thatλ ({T ∈ over(B (S pn ), ̃) : frac({norm of matrix} T {norm of matrix} HS , n) ≥ c 1 t}) ≤ exp (- c 2 t n kp ) for every t ≥ 1, where k p = min {2, 1 + p / 2}, c 1 , c 2 > 1 are universal constants and λ is the Lebesgue measure. This concentration of mass inside a ball of radius proportional to n follows from an almost constant behaviour of the L q norms (with respect to the Lebesgue measure on over(B (S pn ), ̃)) of the Hilbert-Schmidt operator norm of T. The same concentration result holds for every classical ensembles of matrices like real symmetric matrices, Hermitian matrices, symplectic matrices or antisymmetric Hermitian matrices. The result is sharp when p = 1 and p ≥ 2. © 2006 Elsevier SAS. All rights reserved.

author list (cited authors)

  • Guédon, O., & Paouris, G.

citation count

  • 6

publication date

  • January 2007