Numerical range of a pair of strictly upper triangular matrices (Q2894391)

The paper is concerned with the following numerical range: \(W_Y(X) = \{\text{tr} uXu^{-1}Y^*\mid u\in \mathcal U\}\), where \(\mathcal U\) is the group of unit triangular matrices in~\(\mathbb C^{m\times m}\), and \(X,Y\in \mathbb C^{m\times m}\). The set \(W_Y(X)\) is shown to be either a~point (non necessarily the origin) or the entire plane~\(\mathbb C\). A~characterization is obtained for the case when \(W_Y(X)\) is a~point. Convexity results are obtained for a~similar range~\(W^1_Y(X)\), where \(\mathcal U\)~is replaced by any ball of~\(\mathbb C^k\), \(k=m(m-1)/2\), embedded in~\(\mathcal U\), \(m\leqslant 4\). The convexity of~\(W^1_Y(X)\) for \(m>4\) is unknown.