论文标题
禁止在对角线块上操作的阳性保留者
Positivity preservers forbidden to operate on diagonal blocks
论文作者
论文摘要
从Schur乘积定理开始，哪些函数作用入门可以保留正面的半足以有悠久的历史[Crelle 1911]，这意味着绝对单调的函数（即具有非负系数的功率序列）保持在所有维度的矩阵上的阳性。 Schoenberg和Rudin的著名结果[Duke Math。 J. 1942，1959]显示了相反：没有其他此类功能。 Guillot和Rajaratnam受到现代应用的动机[Trans。阿米尔。数学。 Soc。 [2015年]将进入的阳性保留者分类为所有维度，仅在非对角线条目上起作用。这两个结果处于“相反的末端”，在这两种情况下，保留者都必须绝对单调。 除了上述两个情况以外的每种情况下，我们都会完成对“对角线/主块”的分类，除了指定的“对角线/主块”。 （实际上，我们在更一般的框架中实现了这一点。）这产生了无尺寸的阳性保留器的第一个示例 - 具有某些禁止的主要块 - 并非绝对单调。
The question of which functions acting entrywise preserve positive semidefiniteness has a long history, beginning with the Schur product theorem [Crelle 1911], which implies that absolutely monotonic functions (i.e., power series with nonnegative coefficients) preserve positivity on matrices of all dimensions. A famous result of Schoenberg and of Rudin [Duke Math. J. 1942, 1959] shows the converse: there are no other such functions. Motivated by modern applications, Guillot and Rajaratnam [Trans. Amer. Math. Soc. 2015] classified the entrywise positivity preservers in all dimensions, which act only on the off-diagonal entries. These two results are at "opposite ends", and in both cases the preservers have to be absolutely monotonic. We complete the classification of positivity preservers that act entrywise except on specified "diagonal/principal blocks", in every case other than the two above. (In fact we achieve this in a more general framework.) This yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic.
