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 Hölder 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.

摘要

在本文中，我们充分表征了配备 Schatten 范数的矩阵空间上的对偶映射。我们的方法基于对 Schatten 范数的 Hölder 不等式饱和度的分析。我们在主要结果中证明，对于$磷\in (1,\infty ),$具有 Schatten- p范数的实值矩阵空间上的对偶映射是一个连续的单值函数，并为其计算提供了明确的形式。对于特殊情况p  = 1，映射是设置值的；通过添加秩约束，我们表明它可以简化为一个 Borel 可测量的单值函数，我们还为其提供了一个封闭形式的表达式。