A Note on the Optimal Poincaré Constant in Non-Commutative Probability Space
Syeda Rabab Mudakkar · 2012
Abstract: In classical probability, the discrete Poincaré constant of a random variable relates the variance of a function of variable to the expected square of its finite difference and determines the spectral gap. We consider this idea to identify the optimal Poincaré constant of a positive self-adjoint operator in noncommutative probability space. We first identify the optimal Poincaré constant of a positive self-adjoint element having Bernoulli distribution measure. Then extend it to the 3-point and 4-point distribution measure. Finally, we obtain the result for n-point distribution measure, which can be related to the distribution of an element belongs to the space of (n×n) real Hermitian matrices. Key words: Poincaré inequality • spectral gap • self-adjoint operator • bernoulli measure INTRODUCTION AND PRELIMINARIES Poincaré (or spectral gap) inequalities provide a relationship between L 2-norms on functions and their derivatives. It is quite well known [1-6] and the references therein), that the classical Poincaré