Duality Mapping for Schatten Matrix Norms

TL;DR

This paper characterizes the duality mapping for matrices with Schatten norms, providing explicit formulas and analyzing special cases, which advances understanding of matrix norm dualities.

Contribution

It offers a complete characterization of the duality mapping for Schatten norms, including explicit formulas and handling special cases with rank constraints.

Findings

Duality mapping is continuous and single-valued for p in (1,∞).

Explicit formulas for the duality mapping are provided.

Set-valued mapping at p=1 can be reduced to a measurable single-valued function.

Abstract

In this paper, we fully characterize the duality mapping over the space of matrices that are equipped with Schatten norms. Our approach is based on the analysis of the saturation of the H\"older inequality for Schatten norms. We prove in our main result that, for $p\in (1,\infty)$, the duality mapping over the space of real-valued matrices with Schatten-$p$ norm is a continuous and single-valued function and provide an explicit form for its computation. For the special case $p = 1$, the mapping is set-valued; by adding a rank constraint, we show that it can be reduced to a Borel-measurable single-valued function for which we also provide a closed-form expression.