Hilbert spaces of Cauchy transforms

We consider the spaces of functions which can be represented as the Cauchy integrals with $L^2$ data with respect to some fixed measure in the complex plane. These are Reproducing Kernel Hilbert spaces of functions analytic outside the support of the measure. They naturally appear in functional models for rank one perturbations of normal operators.
In the talk we concentrate on the case of discrete measures and discuss several properties of associated spaces of entire functions, such as:

• Sets of uniqueness;
• Existence of orthogonal bases and Riesz bases of reproducing kernels;
• Structure of nearly invariant (i.e., division invariant) subspaces;
• Existence of nearly invariant subspaces of finite codimension and the domain of multiplication by the independent variable;
• Localization of zeros.