Vol. 15, No. 1, 1965

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ISSN: 0030-8730
Decomposition theorems for Fredholm operators

Theodore William Gamelin

Vol. 15 (1965), No. 1, 97–106
Abstract

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
Milestones
Received: 10 February 1964
Published: 1 March 1965
Authors
Theodore William Gamelin