27th Conference of the International Linear Algebra Society (ILAS 2026)

On a characterization of the equality case in the generalized Böttcher-Wenzel inequality

May 18, 2026, 11:25 AM

25m

Goodwin Hall 145

Minisymposium Talk Matrix Inequalities, Matrix Equations, and Their Applications Matrix Inequalities, Matrix Equations, and Their Applications

Speaker

Motoyuki NOBORI (Graduate School of Science and Engineering, Ehime University)

Description

The main purpose of this presentation is to illustrate an equivalent condition for the equality case in the generalized Böttcher-Wenzel (BW) inequality for three matrices.
Let $A, B,$ and $C$ be square matrices with complex elements. The BW inequality is an upper bound estimate on the Frobenius norm of the commutator of $A$ and $B$, defined as $AB-BA$. After the BW inequality was proved in 2008, various subsequent problems related to it have been considered until now. An necessary and sufficient condition for the equality case in the inequality is one of them. Also, there are some papers which investigate relationships between the BW inequality and the uncertainty principle in quantum physics. Recently, the presenter estimated the Frobenius norm of the generalized commutator of $A, B$, and $C$, defined as $ABC-CBA$, and obtained an inequality, which is a generalization of the BW inequality [M. Nobori, A generalization of the Böttcher-Wenzel inequality for three rectangular matrices, Linear Algebra Appl. 725 (2025) 135–144]. In this talk, we shall deduce an equivalent condition for the equality case in the generalized BW inequality. The core idea of the derivation is a necessary and sufficient condition for the Weyl inequality.