Semi-Fredholm operator

Let $X$ and $Y$ be two Banach spaces and let $L(X,Y)$ denote the Banach space of all continuous (bounded) operators from $X$ into $Y$ (cf. also Banach space; Continuous operator). For an operator $T$ in $L(X,Y)$, let $\ker T$ be the set of all $x \in X$ such that $T x = 0$ and let $\coker T$ be the quotient space $Y/TX$, where $TX$ denotes the range of $T$. By definition, $T$ is a semi-Fredholm operator if $TX$ is closed (i.e. it is a normally-solvable operator) and at least one of the vector spaces $\ker$ and $\coker T$ is of finite dimension. (The definition is partially redundant, since if the dimension of $\coker T$ is finite, $TX$ is closed.)

For a semi-Fredholm operator $T$, its index, i.e.

$$ \dim \ker T - \dim \coker T, $$

is uniquely determined either as an integer, or as plus or minus infinity. In the first case $T$ is a Fredholm operator. Cf. also Index of an operator.

The set $SF(X,Y)$ of all semi-Fredholm operators in $L(X,Y)$ is open in $L(X,Y)$ and the index is constant on each connected component of $SF(X,Y)$. Moreover, if $K$ is a compact operator in $L(X,Y)$ and $T$ is in $SF(X,Y)$, then $T+K$ is also in $SF(X,Y)$ and its index equals that of $T$. Due to these properties, semi-Fredholm operators play an important role in linear and non-linear analysis. They were first explicitly considered by I.C. Gohberg and M.G. Krein [a1] and T. Kato [a2], who also treated the case when $T$ is unbounded.

References

[a1]	I.C. Gohberg, M.G. Krein, "The basic propositions on defect numbers, root numbers and indices of linear operators" Transl. Amer. Math. Soc. (2) , 13 (1960) pp. 185–264 Uspekhi Mat. Nauk. , 12 (1957) pp. 43–118
[a2]	T. Kato, "Perturbation theory for nullity, deficiency and other quantities of linear operators" J. d'Anal. Math. , 6 (1958) pp. 261–322