Introducing Discrepancy Values of Matrices and their Application to Bounding Norms of Commutators

We introduce discrepancy values, quantities that are inspired by the notion of spectral spread of Hermitian matrices. In particular, the discrepancy values capture the difference between two consecutive (Ky-Fan-like) pseudo-norms that we also introduce. As a result, discrepancy values share many properties with singular values and eigenvalues, and yet are substantially different to merit their own study. We describe several key properties of discrepancy values and establish a set of useful tools (e.g., representation theorems, majorization inequalities, convex optimization formulations, etc.) for working with them. As an important application, we illustrate the role of discrepancy values in deriving tight bounds on the norms of commutators.