Decomposition theorems for Fredholm operators
Theodore W. Gamelin · Pacific Journal of Mathematics · 1965
This paper is devoted to proving and discussing several consequences of the following decomposition theorem:Let A and B be closed densely-defined linear operators from the Banach space X to the Banach space Y such that D(B) 2 D(A), D(B*) 2 JD(A*), the range R(A) of A is closed, and the dimension of the null-space N(A) of A is finite.Then X and Y can be decomposed into direct sums X = X o Θ Xu Y = γ 0 0 γ l9 where X ί and Y x are finite dimensional, X λ g D(A), X o nD(A) is dense in X, and (X o , Y o ) and (X ί9 Yi) are invariant pairs of subspaces for both A and 5. Let A* and Bi be the restrictions of A and J5 respectively to Xu For all integers ft, (tfoAΓXO) £ #(A), and dim (BoAϊ^CO) =fc dim (EoA^XO) = ft dim #(A 0 ) .Also, the action of Ai and Bι from Xi to Yi can be given a certain canonical description.The object of this paper is to study the operator equation Ax -XBx =y, where A and B are (unbounded) linear operators from a Banach space X to a Banach space Y.In §1, an integer μ(A:B) is defined, which expresses a certain interrelationship between the null space of A and the null space of B. In §1 and 2, decomposition theorems are proved which refine theorem 4 of [2].The theorems allow us to split off certain finite dimensional invariant pairs of subspaces of X and Y so that A and B are well-behaved with respect to μ(A:B) on the remainder.In §4, the stability of these decompositions under perturbation of A by XB is investigated.In § 5, relations between the dimensions of certain subspaces of X and Y are given, and a formula for the Fredholm index of A -XB is obtained.These extend results of Kaniel and Schechter [1], who consider the case X= Y and B the indentity operator.It should be noted that the results of Kaniel and Schechter referred to here follow from theorems 3 and 4 of [2].The results of this paper properly refine Kato's results only when the null space of B is not {0}.1. We will be considering linear operators T defined on a dense linear subset D(A) of a Banach space X 9 and with values in a Banach space Y. N(T) and R(T) will denote the null space and range of T respectively, while a{T) is the dimension of N{T), and β(T) is the