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