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

Abstract

Let \(D\) be positive definite Hermitian and let \(T_X:Y\mapsto XDY-YDX\) be the associated twisted commutator on \(M_n(\mathbb C)\) with the Frobenius norm. For a rank-one factor \(X=\sigma uv^*\), an orthogonal block decomposition gives the exact operator norm \[\lVert T_X\rVert_{F\to F}^2=\sigma^2\bigl(\lVert Du\rVert^2+\lVert Dv\rVert^2-|u^*Dv|^2\bigr),\] as well as the complete singular-value spectrum of \(T_X\). We deduce its rank, nullity, and all Schatten norms. We characterize the factors that attain the associated sharp bound, identify their maximal singular vectors, and prove two quantitative stability estimates. The substitution \(D=\omega^{-1/2}\) transfers these results to the weighted Frobenius norm \(\lVert A\rVert_\omega^2=\operatorname{tr}(A^*A\omega)\). For a fixed rank-one matrix \(A=ab^*\), it yields the exact norm of \(B\mapsto[A,B]\). Consequently, the weighted B\"ottcher\textendash Wenzel inequality of type~(i) holds whenever one factor has rank one, with no normality assumption. This extends the theorem of Fang and Cheng, which requires both factors to have rank one. We also obtain a sharp bound for factors supported on an invariant plane, two rank-two equality families, and a quantitative estimate for almost invariant planes. The rank-two case remains open. We prove exact reductions rather than a solution. They include an orthogonal energy identity, a deficit decomposition, and a rescaling identity leading to a one-parameter operator pencil. On the extremal rank-one boundary, the interference vanishes and a Schur-complement expansion produces a two-dimensional first-order tangency operator. We compute this operator in two opposite geometric configurations. Any singular vector able to violate the conjectured bound is confined between two subspaces of dimension at most four.