Norm bounds for summation of two normal matrices

AbstractA sharp upper bound is obtained for ∥A+iB∥, where A and B are n×n Hermitian matrices satisfying a1I⩽A⩽a2I and b1I⩽B⩽b2I. Similarly, an optimal bound is obtained for ∥U+V∥, where U and V are n×n unitary matrices with any specified spectra; the study leads to some surprising phenomena of discontinuity concerning the spectral variation of unitary matrices. Moreover, it is proven that for two (non-commuting) normal matrices A and B with spectra σ(A) and σ(B), the optimal norm bound for A+B equalsminλ∈Cmaxα∈σ(A)|α+λ|+maxβ∈σ(B)|β−λ|.Extensionsof the results to infinite dimensional cases are also considered