On the sign patterns of entrywise positivity preservers in fixed dimension
On the sign patterns of entrywise positivity preservers in fixed dimension
复制标题
固定维度中条目正性保持者的符号模式
DOI：
10.1353/ajm.2021.0049
复制
发表时间：
2017
影响因子：
1.7
通讯作者：
T. Tao
中科院分区：
文献类型：
--
作者：
A. Khare;T. Tao
摘要：给定域 $I\subset\Bbb{C}$ 和整数 $N>0$，函数 $f: I\to\Bbb{C}$ 被认为是在正半定 $N\times N$ 矩阵 $A=(a_{jk})\in I^{N\times N}$ 上{\it逐项正性保持}，如果逐项应用对于所有这样的 $A$，$f$ 到 $A$ 的 $f[A]=(f(a_{jk}))$ 是半正定的。勋伯格和鲁丁将所有维度的此类保存器归类为绝对单调的。在固定维度 $N$ 中，类似于 Horn 和 Loewner 的工作的结果表明，任何正性保持器 $f$ 的前 $N$ 非零麦克劳林系数都是正的；如果 $I$ 无界，则最后的 $N$ 系数也为正。然而，人们对高阶系数知之甚少：迄今为止，无界域 $I$ 的唯一例子是绝对单调的，因此在所有维度上都有效；对于有界$I$，非绝对单调保持器的例子很少（而且是最近的）。在本文中，我们提供了固定维度$N$、有界和无界域$I=(0,\rho)$中正性保持器高阶麦克劳林系数符号模式的完整表征。特别地，这表明上述Horn-Loewner型条件无法得到改善。作为进一步的特殊情况，这提供了多项式的第一个示例，该多项式在 $I^{N\times N}$ 中保留正半定矩阵的正性，但在 $I^{(N+1)\times (N+1)}$ 中则不然。我们在这方面的主要工具是柯西-比奈公式以及舒尔多项式的上下界。我们还使用 Harish-Chandra--Itzykson--Zuber 公式代替 Schur 多项式的界限，获得了实指数的类似结果。然后，我们从定性存在界限（足以理解所有可能的符号模式）到精确的定量界限。这是使用 Lam、Postnikov 和 Pylyavskyy 的 Schur 正性结果实现的，特别是提供了整数和实数幂元组阈值界限存在的第二个证明。作为一种应用，我们扩展了之前的定性和定量结果，以了解固定维度中完全非负性的保持者——包括它们的符号模式。我们推导出了几个进一步的应用，包括扩展 Cuttler、Greene 和 Skandera 的 Schur 多项式猜想，以获得实元组弱多数化的新特征。
abstract:Given a domain $I\subset\Bbb{C}$ and an integer $N>0$, a function $f: I\to\Bbb{C}$ is said to be {\it entrywise positivity preserving} on positive semidefinite $N\times N$ matrices $A=(a_{jk})\in I^{N\times N}$, if the entrywise application $f[A]=(f(a_{jk}))$ of $f$ to $A$ is positive semidefinite for all such $A$. Such preservers in all dimensions have been classified by Schoenberg and Rudin as being absolutely monotonic. In fixed dimension $N$, results akin to work of Horn and Loewner show that the first $N$ non-zero Maclaurin coefficients of any positivity preserver $f$ are positive; and the last $N$ coefficients are also positive if $I$ is unbounded. However, very little was known about the higher-order coefficients: the only examples to date for unbounded domains $I$ were absolutely monotonic, hence work in all dimensions; and for bounded $I$ examples of non-absolutely monotonic preservers were very few (and recent).In this paper, we provide a complete characterization of the sign patterns of the higher-order Maclaurin coefficients of positivity preservers in fixed dimension $N$, over bounded and unbounded domains $I=(0,\rho)$. In particular, this shows that the above Horn--Loewner-type conditions cannot be improved upon. As a further special case, this provides the first examples of polynomials which preserve positivity on positive semidefinite matrices in $I^{N\times N}$ but not in $I^{(N+1)\times (N+1)}$. Our main tools in this regard are the Cauchy--Binet formula and lower and upper bounds on Schur polynomials. We also obtain analogous results for real exponents, using the Harish-Chandra--Itzykson--Zuber formula in place of bounds on Schur polynomials.We then go from qualitative existence bounds---which suffice to understand all possible sign patterns---to exact quantitative bounds. This is achieved using a Schur positivity result due to Lam, Postnikov, and Pylyavskyy, and in particular provides a second proof of the existence of threshold bounds for tuples of integer and real powers. As an application, we extend our previous qualitative and quantitative results to understand preservers of total non-negativity in fixed dimension---including their sign patterns. We deduce several further applications, including extending a Schur polynomial conjecture of Cuttler, Greene, and Skandera to obtain a novel characterization of weak majorization for real tuples.
总和、Hadamard 产品和 Hadamard 幂的总正性：结果和反例
影响因子：
1.1
作者：
Fallat S
通讯作者：
Fallat S