Symmetries of Gaussian measures and operator colligations

TL;DR

This paper explores the structure of symmetries in infinite-dimensional Gaussian measures, revealing how the closure of the general linear group relates to operator colligations and their explicit action via polymorphisms.

Contribution

It establishes a connection between the closure of $GL()$ in polymorphisms and a semigroup of operator colligations, providing explicit formulas for their actions.

Findings

Closure of $GL()$ contains a semigroup of operator colligations

Explicit formulas for operator colligation actions on Gaussian measure spaces

Characterization of 'spreading' maps as polymorphisms

Abstract

Consider an infinite-dimensional linear space equipped with a Gaussian measure and the group $GL(\infty)$ of linear transformations that send the measure to equivalent one. Limit points of $GL(\infty)$ can be regarded as 'spreading' maps (polymorphisms). We show that the closure of $GL(\infty)$ in the semigroup of polymorphisms contains a certain semigroup of operator colligations and write explicit formulas for action of operator colligations by polymorphisms of the space with Gaussian measure.