Duality Mapping for Schatten Matrix Norms

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 Holder inequality for Schatten norms. We prove in our main result that, for p is an element of (1, infinity), 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.

Automated summary

This paper fully characterizes the duality mapping over matrix spaces equipped with Schatten norms, proving its continuity and single-valuedness for real-valued matrices with Schatten-p norm. The mapping becomes set-valued for the special case of p = 1 but can be reduced to a Borel-measurable single-valued function with a closed-form expression by adding a rank constraint.