# Noetherian integral equation

An integral equation for which Noether's theorems (see below) are valid. Let $X$ be a Banach space, $A$ a bounded linear operator (mapping) taking $X$ into itself: $A : X \rightarrow X$, and let $A ^ {*}$ be the adjoint operator of $A$. Let

$$\tag{1 } A x = y$$

be a linear equation, where $x$ is an unknown and $y$ is a given element of $X$. Next, let $R ( A)$ be the collection of all $y \in X$ for which (1) is solvable (the range of $A$), and let $N ( A)$ be the collection of all solutions of the corresponding homogeneous equation

$$\tag{2 } A x = 0$$

(the null space or kernel of $A$). The mapping $A$( the equation (1)) is called a Noetherian operator (a Noetherian equation) if the following conditions are satisfied.

1) The operator $A$ defined on the whole Banach space (the equation (1)) is normally solvable, that is, (1) is solvable if and only if the right-hand side $y$ is orthogonal to the solutions of the adjoint homogeneous equation

$$\tag{3 } A ^ {*} \phi = 0 ,\ \ \phi \in X ^ {*} ,$$

that is, $( \phi , y ) = 0$ for every $\phi \in N ( A ^ {*} )$.

2) The homogeneous equations (2) and (3) have only finitely many linearly independent solutions; the number

$$\kappa _ {A} = k - k ^ {*} ,$$

where $k = \mathop{\rm dim} N ( A)$, $k ^ {*} = \mathop{\rm dim} N ( A ^ {*} )$, is called the index of the operator (the index of equation (1)).

A Noetherian operator of index zero is called an (abstract) Fredholm operator and the corresponding equation (1) is called a Fredholm equation. For example, if $V$ is a completely-continuous operator, $V : X \rightarrow X$, then

$$\tag{4 } x - V x = y$$

is a Fredholm equation. It is called a canonical Fredholm equation. If in (4) the completely-continuous mapping on some space is an integral operator,

$$( V x ) ( s) = \int\limits K ( s , t ) x ( t) d t ,$$

then the equation is called a Fredholm integral equation. Similarly, if in the Noetherian equation (1) the linear mapping is given by means of integral operators, then it is called a Noetherian integral equation.

F. Noether  considered integral equations with a Hilbert kernel,

$$\tag{5 } ( A \phi ) ( s) = a ( s) \phi ( s) + \frac{b ( s) } \pi \int\limits _ {- \pi } ^ \pi \mathop{\rm cot} \ \frac{t - s }{2} \phi ( t) d t +$$

$$+ \int\limits _ {- \pi } ^ \pi K ( s , t ) \phi ( t) d t = f ( t) ,$$

where the improper integral is understood in the sense of the principal value. For (5) he established the validity of three theorems, which are nowadays called Noether's theorems. (It is assumed that the data and the unknown function are real and Hölder continuous and that $a ^ {2} ( s) + b ^ {2} ( s) \neq 0$.) These are:

1) the equation is normally solvable:

2) the index of the equation is finite;

3) the index can be computed by the formula

$$\kappa _ {A} = \frac{1} \pi [ \mathop{\rm arg} ( a - i b ) ] _ {- \pi } ^ \pi ,$$

where $[ \cdot ] _ {- \pi } ^ \pi$ denotes the increment of the function between brackets.

Theorem (3) was the first to indicate the existence of adjoint linear integral equations that have a different number of linearly independent solutions. Moreover, it follows from this theorem that the index of (5) does not depend on its completely-continuous part.

A Noetherian operator is sometimes called a Fredholm, a generalized Fredholm, a $\Phi$- operator, or an $F$- operator.

How to Cite This Entry:
Noetherian integral equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noetherian_integral_equation&oldid=47977
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article