Principal Minor Assignment, Isometries of Hilbert Spaces, Volumes of Parallelepipeds and Rescalling of Sesqui-holomorphic Functions

TL;DR

This paper characterizes rescaling relations between functions of two variables on topological and complex domains, extending known finite matrix results to infinite cases, and applies this to describe isometries of Hilbert spaces via volume preservation.

Contribution

It extends the characterization of rescaling relations from finite matrices to infinite sets and applies this to describe Hilbert space isometries through volume preservation.

Findings

Rescaling criteria for functions on topological spaces.

Extension of principal minor equality to infinite sets.

Characterization of Hilbert space isometries via parallelepiped volumes.

Abstract

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.