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.

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: $ 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.

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 $ 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

Comments

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 $.