A relation between diagonal and unconditional basis constants

Abstract

This note establishes an inequality relating the weakly nuclear norm of the identity on a finite dimensional space with the diagonal symmetry constant of the space. An application shows the non-existance of"good" diagonally symmetric bases in L(l~), the space of linear operators on n-dimensional Hilbert space. More precisely, for any basis (bi)i _n t/2 It u 11/18. In addition it is shown that there are spaces with diagonal symmetry constant one, but with unconditional basis constant greater than one.