Geometric interpolation in symmetrically-normed ideals (Q932150)

The author applies the complex interpolation method to norms of \(n\)-tuples of operators in a symmetrically normed ideal \(\mathcal{J}_\phi\subseteq B(\mathcal{H})\) (\(\mathcal{H}\) complex separable Hilbert space) defined by a symmetric norming function \(\phi\). The norms considered define Finsler metrics in a manifold of positive and invertible operators, and can be regarded as weighted \(\phi\)-norms, the weight being a positive invertible operator \(\gamma_{a,b}(t)=a^{1/2}(a^{-1/2}ba^{-1/2})^ta^{1/2}\). He also shows that \(\gamma_{a,b}\) is the shortest curve joining \(a\) and \(b\).