Let H(U) be the set of all holomorphic functions on the unit ball U of the complex plane. The Hardy space H2 is the set of all functions f(z)= that belongs to H(U) such that < ¥. Let j be a holomorphic self map of U. The composition operator Cj on H2 is defined as follows:
Cjf = foj, for all f Î H2
The main purpose of this work can be divided in to three aspects.First, a study of the existence and uniqueness of the solution for special types of linear operator equations, namely the Lyapunov equation.
Second, a discussion of the range for the quaii-Jordan*-derivation.Third, some special types of Lyapunov equation, namely stein equation.
The aim of this thesis is to prove the existence and uniqueness of the mild solutions of semilinear initial value control problems in a suitable Banach spaces as well as their controllability. Some theorems regarding controllability, local and global existence as well as uniqueness of the mild solution in infinite dimensional spaces have been developed in suitable Banach space using the Schauder fixed point theorem and the semigroup theory (compact semigroup).
