Semi-Fredholm operator

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

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

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

The set of all semi-Fredholm operators in is open in and the index is constant on each connected component of . Moreover, if is a compact operator in and is in , then is also in and its index equals that of . 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 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