Decomposition theorems for Fredholm operators

Theodore William Gamelin

Vol. 15 (1965), No. 1, 97–106

DOI: 10.2140/pjm.1965.15.97

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 $D\left(B\right)\supseteq D\left(A\right)$, $D\left(B^{\ast }\right)\supseteq D\left(A^{\ast }\right)$, the range $R\left(A\right)$ of $A$ is closed, and the dimension of the null-space $N\left(A\right)$ of $A$ is finite. Then $X$ and $Y$ can be decomposed into direct sums $X=X_0\oplus X_1$, $Y=Y_0\oplus Y_1$, where $X_1$ and $Y_1$ are finite dimensional, $X_1\subseteq D\left(A\right)$, $X_0\cap D\left(A\right)$ is dense in $X$, and $\left(X_0,Y_0\right)$ and $\left(X_1,Y_1\right)$ are invariant pairs of subspaces for both $A$ and $B$. Let $A_i$ and $B_i$ be the restrictions of $A$ and $B$ respectively to $X_i$. For au integers $k$, ${\left(B_0{A}_0^{-1}\right)}^k\left(0\right)\subseteq R\left(A_0\right)$, and

$$dim{\left(B_0{A}_0^{-1}\right)}^k\left(0\right)=kdim\left(B_0{A}_0^{-1}\right)\left(0\right)=kdimN\left(A_0\right).$$

Also, the action of $A_1$ and $B_1$ from $X_1$ 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