Principal minor assignment, isometries of Hilbert spaces, volumes of parallelepipeds and rescaling of sesqui-holomorphic functions

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×X→C are rescalings if there are non-vanishing functions f,g on X such that M(x,y)=f(x)g(y)L(x,y), for any x,y∈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 Cn 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 ±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.