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 $\cF$ for $\hil\cong\C^d$ we compute those dual frames $\cG$ of $\cF$ that are optimal perturbations of the canonical dual frame for $\cF$ under certain restrictions on the norms of the elements of $\cG$. On the other hand, for a fixed finite frame $\cF=\{f_j\}_{j\in\In}$ for $\hil$ we compute those invertible operators $V$ such that $V^*V$ is a perturbation of the identity and such that the frame $V\cdot \cF=\{V\,f_j\}_{j\in\In}$ - which is equivalent to $\cF$ - is optimal among such perturbations of $\cF$. 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.