Geometric interpolation in p-Schatten class

The aim of this work is to apply the complex interpolation method to norms of n-tuples of operators in the p-Schatten classes of a Hilbert space H. The norms considered define Finsler metrics in a certain manifold of positive operators, and can be regarded as weighted p-norms, the weight being a positive invertible operator. As a by-product, we obtain Clarkson's type inequalities.