# Fredholm kernel

A Fredholm kernel is a function $K ( x, y)$ defined on $\Omega \times \Omega$ giving rise to a completely-continuous operator

$$\tag{* } K \phi \equiv \ \int\limits _ \Omega K ( x, y) \phi ( y) \ dy: E \rightarrow E _ {1} ,$$

where $\Omega$ is a measurable set in an $n$- dimensional Euclidean space, and $E$ and $E _ {1}$ are function spaces. The operator (*) is called a Fredholm integral operator from $E$ into $E _ {1}$. An important class of Fredholm kernels is that of the measurable functions $K ( x, y)$ on $\Omega \times \Omega$ for which

$$\int\limits _ \Omega \int\limits _ \Omega | K ( x, y) | ^ {2} \ dx dy < + \infty .$$

A Fredholm kernel that satisfies this condition is also called an $L _ {2}$- kernel.

A Fredholm kernel is called degenerate if it can be represented as the sum of a product of functions of $x$ alone by functions of $y$ alone:

$$K ( x, y) = \ \sum _ {k = 1 } ^ { m } \alpha _ {k} ( x) \beta _ {k} ( y).$$

If $K ( x, y) = K ( y, x)$ for almost-all $( x, y) \in \Omega \times \Omega$, then the Fredholm kernel is called symmetric, and if $K ( x , y ) = \overline{ {K ( y, x) }}\;$, it is called Hermitian (here the bar denotes complex conjugation). A Fredholm kernel $K ( x, y)$ is called skew-Hermitian if $\overline{ {K ( x, y) }}\; = - K ( y, x)$.

The Fredholm kernels $K ( x, y)$ and $K ( y, x)$ are called transposed or allied, and the kernels $K ( x, y)$ and $\overline{ {K ( y, x) }}\;$ are called adjoint.

## Contents

How to Cite This Entry:
Fredholm kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fredholm_kernel&oldid=46979
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article