Fredholm kernel

A Fredholm kernel is a function defined on giving rise to a completely-continuous operator

(*)

where is a measurable set in an -dimensional Euclidean space, and and are function spaces. The operator (*) is called a Fredholm integral operator from into . An important class of Fredholm kernels is that of the measurable functions on for which

A Fredholm kernel that satisfies this condition is also called an -kernel.

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

If for almost-all , then the Fredholm kernel is called symmetric, and if , it is called Hermitian (here the bar denotes complex conjugation). A Fredholm kernel is called skew-Hermitian if .

The Fredholm kernels and are called transposed or allied, and the kernels and are called adjoint.

References

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

Comments

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: . Hermiticity and skew-Hermiticity are then properties of complex-valued kernels. However, the terminology in the literature varies wildly.

About the terminology allied (transposed) and adjoint see also (the editorial comments to) Fredholm theorems.

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

where is a summable sequence of numbers, and and are sequences of elements in some complete convex circled bounded sets in and , respectively. Suppose that is the dual (cf. Adjoint space) of a locally convex space . Then a Fredholm kernel gives rise to a Fredholm operator of the form

where is the value of the functional at the element . If and are Banach spaces, then every element of 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

Comments

A set in a topological vector space over a normal field is called circled (or balanced) if for all in .