Parrott’s Theorem and Bounded Solutions of a System of Operator Equations

Abstract

As an application of the theory of reproducing kernel Hilbert spaces we prove a theorem on the existence of bounded solutions of a system of linear operator equations in Hilbert spaces. This result is obtained by using locally convex spaces and is a corollary to an extension of the well-known Parrott’s theorem. Our theorems extend the Strong Parrott Theorem and its extensions. Our tools for the proof are the generalized reproducing kernel Hilbert spaces due to Schwartz, and the complementary spaces due to de Branges and Rovnyak.