Clarkson-McCarthy Interpolated Inequalities in Finsler norms

Abstract

We apply the complex interpolation method to prove that, given two spaces B (n) p0,a;s0 , B(n) p1,b;s1 of n-tuples of operators in the p-Schatten class of a Hilbert space H, endowed with weighted norms associated to positive and invertible operators a and b of B(H) then, the curve of interpolation (B (n) p0,a;s0 , B(n) p1,b;s1 )[t] of the pair is given by the space of n-tuples of operators in the pt-Schatten class of H, with the weighted norm associated to the positive invertible element γa,b(t) = a 1/2 (a −1/2 ba−1/2 ) ta 1/2 .