On models of function type for a special class of normal operators in Krein spaces and their polar representation

Abstract

The paper is devoted to a function model representation of a normal operator $N$ acting in a Krein space. We assume that $N$ and its adjoint operator $N^{\#}$ have a common invariant subspace $L_{+}$ which is a maximal nonnegative subspace and has a representation as a sum of a finite-dimensional neutral subspace and a uniformly positive subspace. For $N$ we construct a model representation as the multiplication operator by a scalar function acting in a suitable function space. This representation is applied to the problem of existence of a polar representation for normal operators of $D_{\kappa}^+$-class.