主子式配置、希尔伯特空间的等距、类椭球的扩张和类全纯函数的再调用
Principal Minor Assignment, Isometries of Hilbert Spaces, Volumes of Parallelepipeds and Rescalling of Sesqui-holomorphic Functions
摘要：
在这篇文章中，我们考虑以下等价关系的类的两个变量的所有功能的一个集合$X$：我们将说，$L,M: X\times X\to \mathbb{C}$是重标度，如果有非零功能$X$ $f,g$，使$M\left(x,y\right)=f\left(x\right)g\left(y\right) L\left(x,y\right)$，任何$x,y\in X$。当$X$是拓扑空间，且$L$和$M$分别连续时，或当$X$是$\mathbb{C}^{n}$中的整环，且$L$和$M$是倍半全纯时，给出了重标度的判别准则.一种特殊的情况是当$L$和$M$对称时，$f=g$只有值$\pm 1$。在$X$是有限的情况下（因此$L$和$M$是方阵），$M$和$L$之间的这种关系已知由这些矩阵的主子式相等来表征。我们将这个结果推广到$X$为无穷大的情况。作为应用，我们将希尔伯特空间在弱连通集上的等距限制刻画为保持有限个向量集合所跨越的平行六面体体积的映射。
英文摘要：
In this article we consider the following equivalence relation on the class of all functions of two variables on a set $X$: we will say that $L,M: X\times X\to \mathbb{C}$ are rescalings if there are non-vanishing functions $f,g$ on $X$ such that $M\left(x,y\right)=f\left(x\right)g\left(y\right) L\left(x,y\right)$, for any $x,y\in X$. We give criteria for being rescalings when $X$ is a topological space, and $L$ and $M$ are separately continuous, or when $X$ is a domain in $\mathbb{C}^{n}$ and $L$ and $M$ are sesqui-holomorphic. A special case of interest is when $L$ and $M$ are symmetric, and $f=g$ only has values $\pm 1$. This relation between $M$ and $L$ in the case when $X$ is finite (and so $L$ and $M$ are square matrices) is known to be characterized by the equality of the principal minors of these matrices. We extend this result to the case when $X$ is infinite. As an application we characterize restrictions of isometries of Hilbert spaces on weakly connected sets as the maps that preserve the volumes of parallelepipeds spanned by finite collections of vectors.
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