Vol. 15, No. 1, 1965

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Decomposition theorems for Fredholm operators

Theodore William Gamelin

Vol. 15 (1965), No. 1, 97–106

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) D(A), D(B) D(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 = X0 X1, Y = Y 0 Y 1, where X1 and Y 1 are finite dimensional, X1 D(A), X0 D(A) is dense in X, and (X0,Y 0) and (X1,Y 1) are invariant pairs of subspaces for both A and B. Let Ai and Bi be the restrictions of A and B respectively to Xi. For au integers k, (B0A01)k(0) R(A0), and

dim(B0A01)k(0) = kdim(B 0A01)(0) = kdimN(A 0).

Also, the action of A1 and B1 from X1 to Y 1 can be given a certain canonical description.

Mathematical Subject Classification
Primary: 47.10
Received: 10 February 1964
Published: 1 March 1965
Theodore William Gamelin