Sharp Concave-Function Transfer for Lee-Type Schatten Norm Inequalities

Abstract

We compare the sharp constants in Lee-type Schatten norm inequalities for finite families of complex matrices with the corresponding constants obtained after applying an arbitrary nonnegative concave function to the relevant absolute values. We prove that the nonlinear problem has exactly the same best constant as the underlying linear problem for every finite matrix dimension, every number of summands, and every Schatten exponent, including the operator-norm endpoint. For finite exponents, the proof proceeds through a lossless single-cap reduction, an exact representation as a positive mixture for finite cap combinations, and a noncommutative reassembly argument based on a weighted Schatten contraction and the Araki-Lieb-Thirring inequality. Finite-spectrum cap interpolation and a perturbation argument at zero then yield the general concave case. Thus every sharp linear result transfers without loss to the full class of nonnegative concave functions.