Unbounded linear operators and ordinary differential equations

Abstract

The object of this thesis is to use the theory of unbounded linear operators in Banach spaces to find the solutions of certain linear ordinary differential equations. The approach taken is based on that of Dunford and Schwartz [2] (Chapter XIII) and Goldberg [1] (Chapter VI). In Chapter I I shall present a number of results on closed linear operators and their adjoints which will be needed later. Chapter II introduces formal differential operators and differential operators on the Lebesgue spaces. Following this, I shall determine the conjugates of these operators and define boundary values associated with them. Finally I shall give a formula for calculating the inverses of these differential operators, whenever they exist. Chapter III deals with the theory of self-adjoint differential operators defined on L2(I). In the case where the operator has a discrete spectrum, solutions of the related differential equations are obtained by considering eigenfunction expansions. In the general case these solutions are found by using the theory of spectral representations.