The one-parameter Böttcher--Wenzel conjecture holds for \(3\times 3\) nilpotent matrices

Abstract

Chru\'sci\'nski, Kimura, Ohno, and Singal conjectured a sharp bound for a one-parameter deformation of the B\"ottcher\textendash Wenzel functional on pairs of complex matrices, with an explicit constant $c(q)$ depending on a real parameter $q$. This paper proves the conjecture for all $A,B\in M_3(\mathbb C)$ such that $A$ is nilpotent, with $B$ arbitrary, and for every real $q$. After a unitary normalisation, the problem is reduced to a spectral bound for an explicit self-adjoint superoperator. A shifted characteristic-polynomial criterion reduces that bound to $147$ nontrivial inequalities in $q$, all certified by exact symbolic computation without floating-point arithmetic. The constant is attained for every rank-one nilpotent $A$, and an example shows that the Frobenius norm of $A$ cannot be replaced by its operator norm.