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Proof of Nobori's generalized Böttcher-Wenzel inequality conjecture

  • Shuo Shi
  • Juan Zhang
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Abstract

Let m,n≥ 2. For m× n complex matrices A, C and an n× m matrix B, Nobori (Linear Algebra Appl. 725 (2025) 135--144) proposed the following conjecture on the generalized Böttcher-Wenzel inequality: |ABC−CBA|F2 ≤ 2|B|22|A|(2),22|C|F2, where |·|F is the Frobenius norm, |·|2 is the spectral norm, and |·|(2),2 is the (2,2)-norm defined by |X|(2),2=√(σ1(X)2+σ2(X)2), in which σ1(X) and σ2(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≥ 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.

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Cite

BibTeXRIS
BibTeX
@misc{shi2026proof,
  title = {{Proof of Nobori's generalized Böttcher-Wenzel inequality conjecture}},
  author = {Shuo Shi and Juan Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23520},
  eprint = {2609.23520},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23520},
}
RIS
TY  - UNPB
TI  - Proof of Nobori's generalized Böttcher-Wenzel inequality conjecture
AU  - Shuo Shi
AU  - Juan Zhang
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23520
UR  - https://arxiv.org/abs/2609.23520
AB  - Let m,n≥ 2. For m× n complex matrices A, C and an n× m matrix B, Nobori (Linear Algebra Appl. 725 (2025) 135--144) proposed the following conjecture on the generalized Böttcher-Wenzel inequality: |ABC−CBA|_F^2 ≤ 2|B|_2^2|A|_(2),2^2|C|_F^2, where |·|_F is the Frobenius norm, |·|_2 is the spectral norm, and |·|_(2),2 is the (2,2)-norm defined by |X|_(2),2=√(σ_1(X)^2+σ_2(X)^2), in which σ_1(X) and σ_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≥ 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.
ER  -