Green矩阵上的逐元正性保持子
Entrywise Positivity Preservers on Green Matrices
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中文总结 AI 辅助
本研究分类了离散Green矩阵上保持半正定性的逐元函数，明确了全非负性及Green结构保持的条件，还刻画了Loewner单调的连续可微函数，且分类无需正则性假设。
中文摘要 AI 辅助
我们对带正参数的离散Green矩阵\boldsymbol{G}(p,q)=(p_{\boldsymbol{\text{min}}(i,j)}q_{\boldsymbol{\text{max}}(i,j)})上保持半正定性的逐元函数进行了分类，不要求所得矩阵保留Green结构。对于任意阶矩阵，保持子为零函数以及形如f(t)=\boldsymbol{\text{int}}_{[0,\boldsymbol{\text{infty}})}t^{\boldsymbol{\text{alpha}}}\boldsymbol{\text{d}}\boldsymbol{\text{mu}}(\boldsymbol{\text{alpha}})的函数，其中\boldsymbol{\text{mu}}是非零有限正测度，且积分对所有t>0有限。若要求所得矩阵为全非负矩阵，则非零保持子简化为f(t)=ct^{\boldsymbol{\text{alpha}}}，其中c>0且\boldsymbol{\text{alpha}}\boldsymbol{\text{ge}}0。这些幂函数还保持半正定Green结构，而严格Green结构当且仅当\boldsymbol{\text{alpha}}>0时被保持。这些分类无需任何正则性假设。我们还刻画了在每个固定q的Green族上逐元Loewner单调的连续可微函数：当且仅当f'是非负实幂的正混合（允许零测度）时成立。
英文摘要
We classify the entrywise functions that preserve positive semidefiniteness on discrete Green matrices \(G(p,q)=(p_{\min(i,j)}q_{\max(i,j)})\) with positive parameters, without requiring the resulting matrix to retain Green structure. For matrices of all orders, the preservers are the zero function and the functions \(f(t)=\int_{[0,\infty)}t^α\,dμ(α)\), where \(μ\) is a nonzero finite positive measure and the integral is finite for every \(t>0\). Requiring the resulting matrix to be totally nonnegative reduces the nonzero preservers to \(f(t)=ct^α\), where \(c>0\) and \(α\ge0\). These power functions also preserve positive semidefinite Green structure, while strict Green structure is preserved precisely when \(α>0\). No regularity assumption is needed for these classifications. We also characterize continuously differentiable functions that are entrywise Loewner monotone on every fixed-\(q\) Green family: this holds precisely when \(f'\) is a positive mixture of nonnegative real powers, with the zero measure allowed.
