Distance and best approximations in operator norm and trace class norm

TL;DR

This paper investigates the problems of best approximation and distance measurement in operator and trace class spaces, providing formulations and computational insights, especially for finite-dimensional $C^*$-algebras.

Contribution

It introduces new formulations for distance problems in operator and trace class spaces, including finite-dimensional $C^*$-algebras, with computational advantages demonstrated through examples.

Findings

Formulated distance problems in operator and trace class spaces.

Provided computational methods and advantages for finite-dimensional $C^*$-algebras.

Illustrated results with practical examples.

Abstract

We study the best approximation and distance problems in the operator space $\B(\HS)$ and in the space of trace class operators $\LS^1(\B(\HS))$. Formulations of distances are obtained in both cases. The case of finite-dimensional $C^*$-algebras is also considered. The computational advantage of the results is illustrated through examples.