Multiplicative Lidskii's inequalities and optimal perturbations of frames

In this paper we study two design problems in frame theory: on the one hand, given a fixed finite frame F={fj}j∈InF={fj}j∈In for CdCd we compute those dual frames GG of FF that are optimal perturbations of the canonical dual frame for FF under certain restrictions on the norms of the elements of GG. On the other hand, we compute those V⋅F={Vfj}j∈InV⋅F={Vfj}j∈In – for invertible operators V   which are close to the identity – that are optimal perturbations of FF. That is, we compute the optimal perturbations of FF among frames G={gj}j∈InG={gj}j∈In that have the same linear relations as FF. In both cases, optimality is measured with respect to submajorization of the eigenvalues of the frame operators. Hence, our optimal designs are minimizers of a family of convex potentials that include the frame potential and the mean squared error. The key tool for these results is a multiplicative analogue of Lidskii's inequality in terms of log-majorization and a characterization of the case of equality.