# Fredholm operator

A linear normally-solvable operator acting on a Banach space with index equal to zero . The classic example of a Fredholm operator is an operator of the form

 (1)

where is the identity and is a completely-continuous operator on . In particular, on the spaces or an operator of the form

 (2)

where the kernel is a continuous, respectively square-integrable, function on , is Fredholm.

There are Fredholm operators different from (1) (see [2]). Among them are, under certain conditions, for example, an operator of the form , where 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 with a Fredholm operator (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 on of finite index .

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