# 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 ). 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).

How to Cite This Entry:
Fredholm operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fredholm_operator&oldid=46980
This article was adapted from an original article by A.B. Bakushinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article