Norm inequalities for Cartesian decompositions
Let \(T=A+iB\) be the Cartesian decomposition of a complex matrix \(T\) with \(A,B\) Hermitian. Let \(\alpha_j\), \(\beta_j\) be the eigenvalues of \(A\) and \(B\), respectively, ordered such that \(|\alpha_1|\geqslant \dots \geqslant |\alpha_n|\) and \(|\beta_1|\geqslant \dots \geqslant |\beta_n|\). \textit{T. Ando} and \textit{R. Bhatia} [Linear Multilinear Algebra 22, No. 2, 133-147 (1987; Zbl 0641.15007)] conjectured that \(\|\text{diag}(\alpha_1+i\beta_1,\dots,\alpha_n+i\beta _n)\|\leqslant \sqrt 2\|T\|\) for any unitarily invariant norm \(\|\cdot\|\). Here the author proves the above conjecture.
scientific article; zbMATH DE number 1331739