The ultimate estimate of the upper norm bound for the summation of operators

Let A and B be bounded linear operators acting on a Hilbert space H. It is shown that the triangular inequality serves as the ultimate estimate of the upper norm bound for the sum of two operators in the sense thatsup{∥U*AU+V*BV∥:U and V are unitaries}=min{∥A+μI∥+∥B-μI∥:μ∈C}.sup{∥U*AU+V*BV∥:U and V are unitaries}=min{∥A+μI∥+∥B-μI∥:μ∈C}.Consequences of the result related to spectral sets, the von Neumann inequality, and normal dilations are discussed. Furthermore, it is shown that the above equality can be used to characterize those unitarily invariant norms that are multiples of the operator norm in the finite-dimensional case.