Fredholm mapping

For Banach spaces , (cf. Banach space), let denote the set of bounded linear operators from to with domain (cf. also Linear operator). An operator is called a Fredholm mapping if

1) ;

2) is closed in ;

3) . Here, , denote the null space and range of , respectively.

Properties.

Let , , be Banach spaces. If and , then and

(a1)

where (the index). If and is a compact operator from to , then and

(a2)

Moreover, for each there is a such that and

(a3)

for each bounded mapping from to such that . If , are such that , then implies that and . The same is true if . If , then its adjoint operator is in with , where , denote the dual spaces of , , respectively (cf. also Adjoint space).

If , it follows that for each positive integer if . Let

and

A necessary and sufficient condition for both and to be finite is that there exist an integer and operators and , compact on , such that , where denotes the identity operator.

Semi-Fredholm operators.

Let denote the set of all such that is closed in and . Similarly, is the set of all such that is closed in and . If , then . If then . If and is compact from to , then and . If , then there is a such that , , , and for any such that .

Non-linear Fredholm mappings.

Let , be Banach spaces, and let be an open connected subset of . A continuously Fréchet-differentiable mapping from to (cf. also Fréchet derivative) is Fredholm if for each . Set . It is independent of . If is a diffeomorphism, then . If is a compact operator, then is Fredholm with . A useful extension of the Sard theorem due to S. Smale [a2] states that if , are separable (cf. also Separable space), with , then the critical values of are nowhere dense in (cf. also Nowhere-dense set). It follows from this that if has negative index, then contains no interior points, i.e., if there is an such that , then there are points arbitrarily close to such that has no solution in . Consequently, such equations are not considered well posed if has negative index.

Perturbation theory.

The classes and are stable under various types of perturbations. The set of Fredholm perturbations is the set of those such that whenever . The sets of semi-Fredholm perturbations are defined similarly. As noted, compact operators from to are in and . So are strictly singular operators [a3] (in some spaces they may be non-compact). An operator is in if and only if for all . Similarly, it is in if and only if for all . But if and only if for all . On the other hand, if and only if for all compact operators from to . Also, if and only if for all such . Consequently, if and only if and for all compact operators from to .

Perturbation functions.

There are several known "constants" that determine either the fact that a mapping is Fredholm or limit the size of arbitrary perturbations to keep the sum Fredholm. A well-known constant is due to T. Kato [a4]:

where the infimum is taken over those such that . If and , then with (a3) holding. Other constants are:

. A mapping is in if and only if . Moreover, if and , then with (a3) holding.

, where the infimum is taken over all infinite-dimensional subspaces of . A mapping is in if and only if . Moreover, and imply that with (a3) holding.

, where the supremum is taken over all subspaces having finite codimension. If and , then with (a3) holding as well.

Unbounded Fredholm operators.

A linear operator from to is called Fredholm if it is closed, is dense in and , where is considered a Banach space with norm . Many of the facts that are true for bounded Fredholm mappings are true for such operators. In particular, the perturbation theorems hold. In fact, one can generalize them to include unbounded perturbations. A linear operator from to is called -compact if and for every sequence such that , has a convergent subsequence. If is Fredholm and is -compact, then is Fredholm with the same index. A similar result holds when is -bounded. Thus, if is Fredholm, then there is a such that implies that is Fredholm with (a3) holding for . If and is a densely-defined closed operator from to , then , where , denote the conjugates of , , respectively (cf. also Adjoint operator).

References

[a1]	I.C. Gohberg, M.G. Krein, "The basic propositions on defect numbers, root numbers and indices of linear operators" , Transl. Ser. 2 , 13 , Amer. Math. Soc. (1960) pp. 185–264
[a2]	S. Smale, "An infinite dimensional version of Sard's theorem" Amer. J. Math. , 87 (1965) pp. 861–867
[a3]	S. Goldberg, "Unbounded linear operators" , McGraw-Hill (1966)
[a4]	T. Kato, "Perturbation theory for linear operators" , Springer (1966)
[a5]	M. Schechter, "Basic theory of Fredholm operators" Ann. Scuola Norm. Sup. Pisa , 21 (1967) pp. 361–380
[a6]	M. Schechter, "Riesz operators and Fredholm perturbations" Bull. Amer. Math. Soc. , 74 (1968) pp. 1139–1144
[a7]	M. Schechter, "Principles of functional analysis" , Acad. Press (1971)
[a8]	M.S. Berger, "Nonlinearity and functional analysis" , Acad. Press (1977)