Some characterizations of the trace norm triangle equality
The authors obtain a characterization of the equality \(\|A+B\|_{1}=\|A\|_{1}+\|B\|_{1}\) to hold for trace-class operators \(A\) and \(B\), where \(\|X\|_{1}\) means the trace norm. In particular, the authors show that, for trace-class operators \(A\) and \(B\), \(\|A+B\|_{1}=\|A\|_{1}+\|B\|_{1}\) if and only if \(|A+B|=|A|+|B|\). This is related to a result (Theorem 2.3) in [\textit{T. Ando} and \textit{T. Hayashi}, J. Oper. Theory 58, No. 1, 463--468 (2007; Zbl 1150.47001)]. In the present article, the authors use a rather different technique to prove this characterization. Moreover, a characterization of the equality \(\|A+A^{*}\|_{1}=\|A\|_{1}+\|A^{*}\|_{1}\) is obtained.
scientific article; zbMATH DE number 6475069
scientific article