Symmetries of Gaussian measures and operator colligations

Consider an infinite-dimensional linear space equipped with a Gaussian measure and the group $GL(\mathrm{\infty })$ of linear transformations that send the measure to equivalent one. Limit points of $GL(\mathrm{\infty })$ can be regarded as ‘spreading’ maps (polymorphisms). We show that the closure of $GL(\mathrm{\infty })$ 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.