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 [1] 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.

References

[1]	F. Noether, "Ueber eine Klasse singulärer Integralgleichungen" Math. Ann. , 82 (1921) pp. 42–63
[2]	S.M. Nikol'skii, "Linear equations in linear normed spaces" Izv. Akad. Nauk SSSR Ser. Mat. , 7 : 3 (1943) pp. 147–166 (In Russian)
[3]	F.V. Atkinson, "Normal solvability of linear equations in normal spaces" Mat. Sb. , 28 : 1 (1951) pp. 3–14 (In Russian)
[4]	S.G. Krein, "Linear equations in a Banach space" , Birkhäuser (1982) (Translated from Russian)
[5]	S.N. Krachkovskii, A.S. Dikanskii, "Fredholm operators and their generalizations" Progress in Math. , 10 (1971) pp. 37–72 Itogi Nauk. Mat. Anal. 1968 (1969) pp. 39–71
[6]	I.I. Danilyuk, "Nonregular boundary value problems on the plane" , Moscow (1975) (In Russian)
[7]	Z. Prössdorf, "Einige Klassen singulärer Gleichungen" , Akademie Verlag (1974)

Comments

In modern literature the term "completely continuous" is often replaced by "compact operator47B0647B07compact" . Also the term "Fredholm operator" is generally used for linear operators having a finite index. The class of Fredholm operators includes many important operators and there is an extensive literature on the subject. The index satisfies the logarithmic law $ \kappa _ {AB} = \kappa _ {A} + \kappa _ {B} $. For special classes of Fredholm operators, the index can be related to certain topological notions, such as the winding number of a curve (see also above). A bounded linear operator is Fredholm if and only if it invertible modulo the compact operators, i.e., if and only if it corresponds to an invertible element in the Calkin algebra. Normal solvability (i.e., the property of having closed range) is implied by finiteness of the index.

References

[a1]	B. Booss, "Topologie und Analysis, Einführung in die Atiyah–Singer Indexformel" , Springer (1977)
[a2]	J.B. Conway, "A course in functional analysis" , Springer (1985)
[a3]	I.C. Gohberg, M.G. Krein, "The basic propositions on defect numbers, root numbers and indices of linear operators" Transl. Amer. Math. Soc. (2) , 13 (1960) pp. 185–264 Uspekhi Mat. Nauk , 12 : 2 (1957) pp. 43–118
[a4]	S. Goldberg, "Unbounded linear operators" , McGraw-Hill (1966)
[a5]	T. Kato, "Perturbation theory for linear operators" , Springer (1976)
[a6]	I. [I.Ts. Gokhberg] Gohberg, N. Krupnik, "Einführung in die Theorie der eindimensionalen singulären Integraloperatoren" , Birkhäuser (1979) (Translated from Russian)
[a7]	P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , Integral equations - a reference text , Noordhoff (1975) (Translated from Russian)
[a8]	E. Meister, "Randwertaufgaben der Potentialtheorie" , Teubner (1983)