Conjectured Böttcher–Wenzel-type bound with spectral norm of B

Prove that for all positive integers m,n with m,n ≥ 2, and all matrices A,C ∈ M_{m,n}(C) and B ∈ M_{n,m}(C), the generalized commutator ABC − CBA satisfies the inequality ||ABC − CBA||_F^2 ≤ 2 ||B||_2^2 ||A||_{(2),2}^2 ||C||_F^2, where ||·||_F denotes the Frobenius norm, ||·||_2 denotes the spectral norm, and ||A||_{(2),2} denotes the Ky Fan (2,2)-norm of A.

Background

The paper studies a generalized commutator ABC − CBA for A,C ∈ M_{m,n}(C) and B ∈ M_{n,m}(C), establishing the inequality ||ABC − CBA||F^2 ≤ 2 ||C||_2^2 ||A||{(2),2}^2 ||B||_F^2 (Theorem 1), with a tighter bound when m=1 or n=1. This extends classical Böttcher–Wenzel-type bounds to rectangular matrices and to a product of three matrices.

Motivated by symmetry considerations and numerical evidence, the authors propose an analogous bound where the spectral norm applies to B instead of C. They note the conjectured inequality recovers the known form when m=n and B is the identity, and it is verified when m=1 or n=1 by their earlier theorem. As a consequence of the conjecture, a related Kronecker-product bound would follow, and they also establish results guaranteeing this consequence when rank(B) ≤ 2. A stronger rank-dependent generalization is shown to be false, underscoring the sharpness and subtlety of the conjectured inequality.