秩一算子的ρ-数值半径与参数化Buzano型不等式

中文摘要 AI 辅助

设ℋ为复希尔伯特空间，ρ>0，我们建立了秩一算子的ρ-数值半径的显式公式：ω_ρ(a⊗b)= (1/ρ)||a|| ||b|| + |1 - 1/ρ| |⟨a,b⟩|，其中a,b∈ℋ，该公式是算子范数、数值半径与谱半径的统一表达式，分别对应ρ=1、ρ=2及ρ→∞的情形。作为应用，我们推导了针对四个向量的参数化Buzano型不等式族，从中提取显式闭式界；在实希尔伯特空间自动满足的实条件下，确定了该族中最优闭式界。实例表明所得界严格优于柯西-施瓦茨不等式与Buzano不等式，经典Buzano不等式作为边界情形被导出，为其提供了另一种证明。

英文摘要

Let $\mathcal{H}$ be a complex Hilbert space and $ρ>0$. We establish the explicit formula \[ ω_ρ(a\otimes b)=\frac{1}ρ\|a\| \|b\|+\left|1-\frac{1}ρ\right||\langle a,b\rangle|,\qquad a,b\in\mathcal{H}, \] for the $ρ$-numerical radius of rank-one operators, a unified expression interpolating between the operator norm, the numerical radius, and the spectral radius, which correspond to $ρ=1,2$, and $ρ\to\infty$, respectively. As an application, we derive a parametrized family of Buzano-type inequalities for four vectors. We extract from it an explicit closed-form bound and, under a reality condition which is automatically satisfied in real Hilbert spaces, we determine the best bound in the family in closed form. Examples show that the resulting bounds can be strictly sharper than both the Cauchy--Schwarz and the Buzano inequality, and the classical Buzano inequality is recovered as a boundary case, which yields an alternative proof of it.

发表机构

• Faculty of Electronic Engineering, University of Niš（尼什大学电子工程学院）

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