The ultimate estimate of the upper norm bound for the summation of operators (Q820066)

Let \(A\) and \(B\) be bounded linear operators acting on a Hilbert space \(H\). It is shown that \[ \sup\{\| U^*AU+V^*BV\| : U,V \text{ unitaries}\}= \min\{\| A+\mu I\| +\| B-\mu I\|:\mu\in\mathbb{C}\}.\tag{1} \] Furthermore, it is shown that the above quantity is the same as \[ \sup\{\| AX+XB\|:X\in B(H),\;\| X\| \leq1\}, \] which leads to a proof of Stampfli's well-known result on the norm of generalised derivation. The equality in (1) may not hold if the Hilbert-space operator norm \(\| \cdot\| \) is replaced by other norms. In fact, condition (1) can be used to characterise those unitarily invariant norms on \(M_n\) which are multiples of the operator norm. As a consequence of (1), some results related to spectral sets, the von Neumann inequality, and normal dilations are discussed.