# 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

#### References

 [1] V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) pp. Chapt. 1 (Translated from Russian)

A completely-continuous operator is nowadays usually called a compact operator.

In the main article above, no distinction is made between real-valued and complex-valued kernels. Usually, symmetry is defined for real-valued kernels, as is skew-symmetry: $K ( x , y ) = - K ( y , x )$. Hermiticity and skew-Hermiticity are then properties of complex-valued kernels. However, the terminology in the literature varies wildly.

A Fredholm kernel is a bivalent tensor (cf. Tensor on a vector space) giving rise to a Fredholm operator. Let $E$ and $F$ be locally convex spaces (cf. Locally convex space), and let $E \overline \otimes \; F$ be the completion of the tensor product $E \otimes F$ of these spaces in the inductive topology, that is, in the strongest locally convex topology in which the canonical bilinear mapping $E \times F \rightarrow E \overline \otimes \; F$ is continuous. An element $u \in E \overline \otimes \; F$ is called a Fredholm kernel if it can be represented in the form

$$u = \sum _ {i = 1 } ^ \infty \lambda _ {i} e _ {i} \otimes f _ {i} ,$$

where $\{ \lambda _ {i} \}$ is a summable sequence of numbers, and $\{ e _ {i} \}$ and $\{ f _ {i} \}$ are sequences of elements in some complete convex circled bounded sets in $E$ and $F$, respectively. Suppose that $E$ is the dual (cf. Adjoint space) $G ^ \prime$ of a locally convex space $G$. Then a Fredholm kernel gives rise to a Fredholm operator $A: G \rightarrow F$ of the form

$$x \rightarrow \ \sum _ {i = 1 } ^ \infty \lambda _ {i} \langle x, e _ {i} \rangle f _ {i} ,$$

where $\langle x, e _ {i} \rangle$ is the value of the functional $e _ {i} \in G ^ \prime$ at the element $x \in G$. If $E$ and $F$ are Banach spaces, then every element of $E \overline \otimes \; F$ is a Fredholm kernel.

The concept of a Fredholm kernel can also be generalized to the case of the tensor product of several locally convex spaces. Fredholm kernels and Fredholm operators constitute a natural domain of application of the Fredholm theory.

#### References

 [1] A. Grothendieck, "La théorie de Fredholm" Bull. Amer. Math. Soc. , 84 (1956) pp. 319–384 [2] A. Grothendieck, "Produits tensoriels topologiques et espaces nucleaires" Mem. Amer. Math. Soc. , 5 (1955)

G.L. Litvinov

A set $A$ in a topological vector space $E$ over a normal field $K$ is called circled (or balanced) if $k A \subset A$ for all $| k | \leq 1$ in $K$.