Noncommutative partially convex rational functions

Abstract

Motivated by classical notions of bilinear matrix inequalities (BMIs) and partial convexity, this article investigates partial convexity for noncommutative functions. It is shown that noncommutative rational functions that are partially convex admit novel butterfly-type realizations that necessitate square roots. A strengthening of partial convexity arising in connection with BMIs -xy-convexity -is also considered. A characterization of xy-convex polynomials is given.