Estimate for distance-coefficient of matrices

Abstract

Motivation of this paper is an open problem exposed by B. Beauzamy [l]. Let M be a 3 X 3 matrix and d(M) is the distance to the diagonal algebra. Let α(M) = sup{∥P¹ MP∥:P is a projection in the diagonal algebra} and then call K(M) = d(M)/α(M) the distance-coefficient of M. The following results are obtained: (1) If M’ has two zero-entries apart from its diagonal. then K(M) ≤3/2√2: (2) If M has one zero-entry apart from its diagonal, then K(M) ≤ √41/32; (3) If M is arbitrary, then K(M) √3/2.