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A Fredholm kernel is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f0414401.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f0414402.png" /> giving rise to a [[Completely-continuous operator|completely-continuous operator]]
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f0414403.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f0414404.png" /> is a measurable set in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f0414405.png" />-dimensional Euclidean space, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f0414406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f0414407.png" /> are function spaces. The operator (*) is called a Fredholm integral operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f0414408.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f0414409.png" />. An important class of Fredholm kernels is that of the measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144011.png" /> for which
+
A Fredholm kernel is a function $  K ( x, y) $
 +
defined on  $  \Omega \times \Omega $
 +
giving rise to a [[Completely-continuous operator|completely-continuous operator]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144012.png" /></td> </tr></table>
+
$$ \tag{* }
 +
K \phi  \equiv \
 +
\int\limits _  \Omega
 +
K ( x, y) \phi ( y) \
 +
dy: E  \rightarrow  E _ {1} ,
 +
$$
  
A Fredholm kernel that satisfies this condition is also called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144014.png" />-kernel.
+
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
  
A Fredholm kernel is called degenerate if it can be represented as the sum of a product of functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144015.png" /> alone by functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144016.png" /> alone:
+
$$
 +
\int\limits _  \Omega
 +
\int\limits _  \Omega
 +
| K ( x, y) |  ^ {2} \
 +
dx  dy  < + \infty .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144017.png" /></td> </tr></table>
+
A Fredholm kernel that satisfies this condition is also called an  $  L _ {2} $-
 +
kernel.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144018.png" /> for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144019.png" />, then the Fredholm kernel is called symmetric, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144020.png" />, it is called Hermitian (here the bar denotes complex conjugation). A Fredholm kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144021.png" /> is called skew-Hermitian if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144022.png" />.
+
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:
  
The Fredholm kernels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144024.png" /> are called transposed or allied, and the kernels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144026.png" /> are called adjoint.
+
$$
 +
K ( x, y)  = \
 +
\sum _ {k = 1 } ^ { m }
 +
\alpha _ {k} ( x)
 +
\beta _ {k} ( y).
 +
$$
 +
 
 +
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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  pp. Chapt. 1  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  pp. Chapt. 1  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
A completely-continuous operator is nowadays usually called a [[Compact operator|compact operator]].
 
A completely-continuous operator is nowadays usually called a [[Compact operator|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: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144027.png" />. Hermiticity and skew-Hermiticity are then properties of complex-valued kernels. However, the terminology in the literature varies wildly.
+
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|Fredholm theorems]].
 
About the terminology allied (transposed) and adjoint see also (the editorial comments to) [[Fredholm theorems|Fredholm theorems]].
  
A Fredholm kernel is a bivalent tensor (cf. [[Tensor on a vector space|Tensor on a vector space]]) giving rise to a [[Fredholm-operator(2)|Fredholm operator]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144029.png" /> be locally convex spaces (cf. [[Locally convex space|Locally convex space]]), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144030.png" /> be the completion of the [[Tensor product|tensor product]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144031.png" /> of these spaces in the inductive topology, that is, in the strongest locally convex topology in which the canonical bilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144032.png" /> is continuous. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144033.png" /> is called a Fredholm kernel if it can be represented in the form
+
A Fredholm kernel is a bivalent tensor (cf. [[Tensor on a vector space|Tensor on a vector space]]) giving rise to a [[Fredholm-operator(2)|Fredholm operator]]. Let $  E $
 +
and $  F $
 +
be locally convex spaces (cf. [[Locally convex space|Locally convex space]]), and let $  E \overline \otimes \; F $
 +
be the completion of the [[Tensor product|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144034.png" /></td> </tr></table>
+
$$
 +
= \sum _ {i = 1 } ^  \infty 
 +
\lambda _ {i} e _ {i} \otimes f _ {i} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144035.png" /> is a summable sequence of numbers, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144037.png" /> are sequences of elements in some complete convex circled bounded sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144039.png" />, respectively. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144040.png" /> is the dual (cf. [[Adjoint space|Adjoint space]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144041.png" /> of a locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144042.png" />. Then a Fredholm kernel gives rise to a Fredholm operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144043.png" /> of the form
+
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|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144044.png" /></td> </tr></table>
+
$$
 +
x  \rightarrow \
 +
\sum _ {i = 1 } ^  \infty 
 +
\lambda _ {i} \langle  x, e _ {i} \rangle f _ {i} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144045.png" /> is the value of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144046.png" /> at the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144047.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144049.png" /> are Banach spaces, then every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144050.png" /> is a Fredholm kernel.
+
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.
 
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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,   "La théorie de Fredholm"  ''Bull. Amer. Math. Soc.'' , '''84'''  (1956)  pp. 319–384</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Grothendieck,   "Produits tensoriels topologiques et espaces nucleaires"  ''Mem. Amer. Math. Soc.'' , '''5'''  (1955)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck, "La théorie de Fredholm"  ''Bull. Amer. Math. Soc.'' , '''84'''  (1956)  pp. 319–384 {{ZBL|0073.10101}}</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  A. Grothendieck, "Produits tensoriels topologiques et espaces nucléaires"  ''Mem. Amer. Math. Soc.'' , '''5'''  (1955)</TD></TR>
 +
</table>
  
 
''G.L. Litvinov''
 
''G.L. Litvinov''
  
 
====Comments====
 
====Comments====
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144051.png" /> in a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144052.png" /> over a normal field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144053.png" /> is called circled (or balanced) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144054.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144055.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144056.png" />.
+
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 $.

Latest revision as of 16:56, 18 May 2024


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 Zbl 0073.10101
[2] A. Grothendieck, "Produits tensoriels topologiques et espaces nucléaires" 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 $.

How to Cite This Entry:
Fredholm kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fredholm_kernel&oldid=12278
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article