#### Vol. 15, No. 1, 1965

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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 $D\left(B\right)\supseteq D\left(A\right)$, $D\left({B}^{\ast }\right)\supseteq D\left({A}^{\ast }\right)$, 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.

Primary: 47.10