The class of B-Fredholm linear relations
Let $X$ be a Banach space. A~linear relation $A$ on $X$ is said to be a range space relation if $G(A)\subset X\times X$ is complete under some stronger norm on $G(A)$ and is called quasi-Fredholm of degree $d$ if it is a range space relation, \[k_n(A)=\dim ((R(A^n)\cap N(A))/(R(A^{n+1})\cap N(A)))=0\text{ for all }n\geq d,\] and the subspaces $N(A^d)+R(A)$ and $R(A^d)\cap N(A)$ are complemented in $X$. In the present paper, the authors prove that, if $A$ is a quasi-Fredholm linear relation of degree $d$ on $X$, then there exist two closed subspaces of $X$, $X_1$ and $X_2$, such that $X=X_1\oplus X_2$, $A|_{X_i}(X_i)\subset X_i$, $i=1,2$, $A|_{X_1}$ is a nilpotent linear operator of degree $d$, and $A|_{X_2}$ is a closed linear relation on $X_2$ with $k_i(A|_{X_2})=0$ for all $i\geq 0$. A~linear relation $A\in\mathcal {LR}(X)$ is called B-Fredholm if it is a range space relation and there exists $d\in \mathbb N$ such that $R(A^d)$ is closed and the restriction $A|_{R(A^d)}$ is a Fredholm linear relation. Some properties of B-Fredholm linear relations are obtained. For instance, it is shown that $A$ is B-Fredholm if and only if $A=A_1\oplus A_2$, with $A_1$ a nilpotent operator and $A_2$ a Fredholm relation.
scientific article; zbMATH DE number 6506171
scientific article