Proof of Nobori's generalized Böttcher-Wenzel inequality conjecture

Let $m,n\ge 2$. For $m\times n$ complex matrices $A$, $C$ and an $n\times m$ matrix $B$, Nobori (Linear Algebra Appl. 725 (2025) 135–144) proposed the following conjecture on the generalized Böttcher-Wenzel inequality: $$\mathrm{\parallel }ABC-CBA{\mathrm{\parallel }}_F^2\le 2\mathrm{\parallel }B{\mathrm{\parallel }}_2^2\mathrm{\parallel }A{\mathrm{\parallel }}_{(2),2}^2\mathrm{\parallel }C{\mathrm{\parallel }}_F^2,$$ where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_F$ is the Frobenius norm, $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_2$ is the spectral norm, and $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{(2),2}$ is the $(2,2)$-norm defined by $\mathrm{\parallel }X{\mathrm{\parallel }}_{(2),2}=\sqrt{{\sigma }_1(X{)}^2+{\sigma }_2(X{)}^2}$, in which ${\sigma }_1(X)$ and ${\sigma }_2(X)$ are the largest and second largest singular values of $X$, respectively. This paper proves the conjecture and, as a consequence, establishes the Kronecker product inequality that Nobori derived from it. In addition, we show that for each fixed $m,n\ge 2$, the constant $2$ is sharp. We also give necessary and sufficient conditions for equality, namely, the equality condition in the Böttcher–Wenzel inequality and some matrix identities derived from a fixed singular value decomposition.