Fredholm operator

A linear normally-solvable operator $ B $ acting on a Banach space $ E $ with index $ \chi _ {B} $ equal to zero $ ( \chi _ {B} = \mathop{\rm dim} \mathop{\rm ker} B - \mathop{\rm dim} \mathop{\rm coker} B) $. The classic example of a Fredholm operator is an operator of the form

$$ \tag{1 } B = I + T, $$

where $ I $ is the identity and $ T $ is a completely-continuous operator on $ E $. In particular, on the spaces $ C ( a, b) $ or $ L _ {2} ( a, b) $ an operator of the form

$$ \tag{2 } B \phi = \phi ( x) + \int\limits _ { a } ^ { b } K ( x, s) \phi ( s) ds, $$

where the kernel $ K ( x, s) $ is a continuous, respectively square-integrable, function on $ [ a, b] \times [ a, b] $, is Fredholm.

There are Fredholm operators different from (1) (see [2]). Among them are, under certain conditions, for example, an operator of the form $ I + K $, where $ K $ is an convolution integral operator on the semi-axis or on the whole axis (that is not completely continuous), and many differential operators.

It is easy to state a variety of theorems asserting that one can solve operator equations of the form $ B \phi = f $ with a Fredholm operator $ B $( see Fredholm kernel).

One also comes across other uses of the term "Fredholm operator" . For example, sometimes a Fredholm operator is any bounded linear operator $ B $ on $ E $ of finite index $ \chi _ {B} $.

In the classical theory of linear integral equations, a Fredholm operator is often the actual integral operator in (2).

References

[1]	M.G. Krein, "Linear equations in a Banach space" , Birkhäuser (1982) (Translated from Russian)
[2]	E. Cartan, "Espaces à connexion affine, projective et conforme" Acta. Math. , 48 (1926) pp. 1–42

Comments

In modern literature the phrase "completely-continuous operator" is often replaced by "compact operator" . Also, the term "Fredholm operator" is generally used for linear operators having a finite index. The class of Fredholm operators (occasionally also called $ \Phi $- operators or Noether operators) includes many important operators and there is an extensive literature on the subject. The index satisfies the logarithmic law $ \chi _ {AB} = \chi _ {A} + \chi _ {B} $. For special classes of Fredholm operators, the index can be related to certain topological notions, such as the winding number of a curve. A bounded linear operator is Fredholm if and only if it is invertible modulo the compact operators, i.e. if and only if it corresponds to an invertible element in the Calkin algebra. Normal solvability (i.e. the property of having closed range) is implied by finiteness of the index.

References

[a1]	B. Booss, "Topologie und Analysis, Einführung in die Atiyah–Singer Indexformel" , Springer (1977)
[a2]	J.B. Conway, "A course in functional analysis" , Springer (1985)
[a3]	I.C. [I.C. Gokhberg] 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 : 2 (74) (1957) pp. 43–118
[a4]	S. Goldberg, "Unbounded linear operators" , McGraw-Hill (1966)
[a5]	T. Kato, "Perturbation theory for linear operators" , Springer (1976)