Distance and best approximations in operator norm and trace class norm

TL;DR

The paper develops exact distance formulas for approximating elements in the operator space $\mathcal{B}(\mathcal{H})$ and its trace-class dual $\mathcal{L}^1(\mathcal{B}(\mathcal{H}))$ by finite-dimensional subspaces, showing that the classical distance $\Delta(x)$ to $\mathcal{K}(\mathcal{H})$ is generally only a lower bound for such subspaces. Using duality between $\mathcal{B}(\mathcal{H})$, $\mathcal{K}(\mathcal{H})$, and $\mathcal{L}^1(\mathcal{B}(\mathcal{H}))$ along with $\sigma$-WOT compactness, the authors derive trace-based distance expressions and give necessary and sufficient conditions for best approximations via Hahn–Banach extensions. The results extend to finite-dimensional $C^*$-algebras, where distances reduce to dual trace formulas and explicit computations, with illustrative examples demonstrating computational advantages. These contributions provide practical, computable criteria for best approximation problems in infinite-dimensional operator spaces and in finite-dimensional $C^*$-algebras.

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.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 1.4: Tak, Chapter II
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7: Singer Theorem 1.1, Page 170
• Lemma 1.8: Rudin, Lemma 3.9
• Example 2.1
• proof : Proof of Theorem \ref{['Thm: Oper.']}