三维情形下的平方函数与完全Crouzeix猜想

中文摘要 AI 辅助

我们解决了阶数不超过3的矩阵的完全Crouzeix猜想，还得到了尖锐的列与行平方函数不等式、类正规算子的严格标量界，以及二维和三维中缩放q-数值域的尖锐完全谱常数，另有刚性、稳定性及表示测度结果。

英文摘要

We settle the complete Crouzeix conjecture for matrices of order at most three and prove the Q-algebra case of the completely bounded Clouâtre--Ostermann--Ransford (COR) conjecture for homomorphisms into matrices of order at most three, via sharp abstract column and row estimates. Our approach also establishes the scalar COR conjecture in a stronger form, for homomorphisms with commutative range on Banach algebras with unity satisfying von Neumann's inequality. Under contractivity of the symmetrized map, this result holds on arbitrary Hilbert spaces without initial boundedness assumptions on the homomorphism or the antilinear map. For arbitrary operator algebras, we disprove the complete COR conjecture by an exact three-dimensional example with target matrix order two. We also prove the sharp complete bound for every matrix subalgebra containing the diagonal, in arbitrary matrix order and on arbitrary target Hilbert spaces. We also obtain column and row square-function inequalities with sharp norm bounds, strict scalar bounds for operators similar to normal operators, sharp complete spectral constants for scaled $q$-numerical ranges in dimensions two and three, and rigidity, stability, and representing-measure results.

发表机构

• Umeå University（于默奥大学）
• Jagiellonian University（雅盖隆大学）
• University of Eastern Finland（芬兰东部大学）
• University of Reading（雷丁大学）
• University of Helsinki（赫尔辛基大学）

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