Norm bounds for summation of two normal matrices. (Q1426296)

A sharp bound is obtained for \(\| A+iB\| \), where \(A\) and \(B\) are \(n\times n\) Hermitian matrices satisfying \(a_1I \leq A\leq a_2I\) and \(b_1I \leq B\leq b_2I\). This question was stated e.g. by \textit{R. Bhatia} [Matrix Analysis (1996; Zbl 0863.15001); Problem I.6.18]. Similarly, an optimal bound is obtained for \(\| U+V\| \), where \(U\) and \(V\) are \(n\times n\) unitary matrices with any specific spectra. The study leads to some suprising phenomena of discontinuity concerning the spectral variation of unitary matrices. It is proven that for two (non-commuting) normal matrics \(A\) and \(B\) with spectra \(\sigma (A)\) and~\(\sigma (B)\), the optimal norm bound for \(A+B\) equals \(\min _{\lambda \in {\mathbb C}} \{\max _{\alpha\in \sigma(A)} | \alpha+\lambda| + \max _{\beta\in\sigma(B)} | \beta-\lambda|\} \). Extensions to infinite-dimenisional cases are also considered.