e-mail: sbelyi@trojan.troyst.edu

**Realizable Operator - Valued Functions**

**Abstract**

Realization theory of different classes of operator-valued (matrix) functions as transfer operator-valued functions of linear systems plays an important role in modern operator, system, control, and scattering theories.

Almost all realizations in the modern theory of non-selfadjoint operators and its applications deal with systems (operator colligations) the main operator of which is a bounded linear operator. The case with an unbounded non-selfadjoint operator as a main operator in a corresponding system has not been investigated thoroughly because of a number of essential difficulties usually connected with unbounded.

A class of the inverse Stieltjes operator-valued functions was first
introduced by M.Krein. These functions act on a finite-dimensional Hilbert
space and have well known integral representation. Realization problems
for the class of the inverse Stieltjes operator-valued functions have not
been investigated at all. Our main goal is to extend some of our recent
results to operator-valued inverse Stieltjes functions.