The complete singular-value spectrum of a twisted commutator with a rank-one factor and an application to the weighted Böttcher–Wenzel inequality

Let \(D=D^*>0\) and consider the twisted commutator operator \(T_X:Y\mapsto XDY-YDX\) on \(M_n(\mathbb C)\) equipped with the Frobenius norm. For a rank-one factor \(X=\sigma uv^*\), we obtain an orthogonal block decomposition of \(T_X\). It gives the exact formula\begin{equation*}\left\lVert T_X\right\rVert_{F\to F}^2 =\sigma^2\bigl(\left\lVert Du\right\rVert^2+\left\lVert Dv\right\rVert^2-|u^*Dv|^2\bigr),\end{equation*}and determines the complete singular-value spectrum of \(T_X\). We deduce its rank, nullity, and every Schatten norm. After the substitution \(D=\omega^{-1/2}\), the same decomposition gives the exact norm of the commutator map with a fixed rank-one factor \(A=ab^*\):\begin{equation*} \left\lVert B\mapsto[A,B]\right\rVert_{\omega\to\omega}^2 =(b^*\omega b)(a^*\omega^{-1}a) +\left\lVert a\right\rVert^2\left\lVert b\right\rVert^2-|a^*b|^2.\end{equation*}Consequently, the weighted B"ottcher--Wenzel \ inequality of type~(i) holds whenever one factor has rank one. No normality assumption is required. We characterize the rank-one equality cases and prove quantitative stability estimates. We also treat matrices supported on invariant and almost invariant planes. For rank two, we derive orthogonal energy and deficit decompositions. An exact rescaling identity reformulates the conjecture as the positivity of a one-parameter operator pencil. At an extremal rank-one boundary, a Schur-complement expansion defines a two-dimensional tangency operator. We compute this operator in two exact geometric configurations. It equals \(|u^*Dv|^2I\) when the second singular directions lie in the extremal plane, and \((d_1^2+d_2^2)I\) when both directions are transverse to that plane. We also distinguish common maximizing directions from compensated tangency directions in the reciprocal rank-two equality family. Finally, we prove a localization theorem for singular vectors that could violate the conjectured bound.