Noetherian operator

A linear operator (with closed range) that is simultaneously $ n $- normal and $ d $- normal (see Normally-solvable operator). In other words, a Noetherian operator $ A $ is a normally-solvable operator of finite $ d $- characteristic ( $ n ( A) < + \infty $, $ d ( A) < + \infty $). The index $ \chi ( A) $( cf. Index of an operator) of a Noetherian operator $ A $ is also finite. The simplest example of a Noetherian operator is a linear operator acting from $ \mathbf R ^ {k} $ to $ \mathbf R ^ {l} $. It is named after F. Noether [1], in whose work the theory of Noetherian operators is developed parallel to the theory of singular integral equations. Linear operators generated by general boundary value problems for elliptic equations are frequently Noetherian.

In practice, as a rule one succeeds to verify the validity of the following propositions (Noether's theorems):

1) the equation $ A x = 0 $ has either no non-trivial solutions or a finite number $ n $ of linearly independent solutions; and

2) the inhomogeneous equation $ A x = y $ is either solvable for any right-hand side $ y $, or for its solvability it is necessary and sufficient that $ \langle y , \psi _ {i} \rangle = 0 $, $ i = 0 \dots m $, where $ \{ \psi _ {i} \} _ {0} ^ {m} $ is a complete system of linearly independent solutions of the associated homogeneous equation, or it is formally adjoint to the homogeneous problem.

From 1) and 2) it follows that $ A $ is a Noetherian operator.

The property of being Noetherian is stable: If $ A $ is a Noetherian operator and $ B $ is a linear operator of sufficiently small norm or is completely continuous, then $ A + B $ is also Noetherian, and $ \chi ( A + B ) = \chi ( A) $.

Suppose that $ A \in L ( X , Y ) $, where $ L ( X , Y ) $ is the space of linear operators from $ X $ to $ Y $, is Noetherian. Then there is the direct decomposition

$$ X = N ( A) \dot{+} \widehat{X} ,\ Y = Z \dot{+} R ( A) , $$

where $ N ( A) $ is the null space of $ A $, $ R( A) $ is the range of $ A $ and $ \mathop{\rm dim} Z = d ( A) $. The general solution of the equation $ Ax= y $, $ y \in R ( A) $, is of the form $ x = {\widehat{A} } {} ^ {-} 1 y + v $, where $ \widehat{A} \in \widehat{L} ( \widehat{X} , R ( A) ) $, $ \widehat{A} = A $ on $ \widehat{X} $( the restriction of $ A $) and $ v \in N ( A) $ is arbitrary. If $ A $ is Noetherian with $ d $- characteristic $ ( n , m ) $, then $ A ^ {*} $ is Noetherian with $ d $- characteristic $ ( m , n ) $.

References

[1]	F. Noether, "Ueber eine Klasse singulärer Integralgleichungen" Math. Ann. , 82 (1921) pp. 42–63
[2]	S.G. Krein, "Linear differential equations in Banach space" , Transl. Math. Monogr. , 29 , Amer. Math. Soc. (1971) (Translated from Russian)
[3]	M.M. Vainberg, V.A. Trenogin, "Theory of branching of solutions of non-linear equations" , Noordhoff (1974) (Translated from Russian)

Comments

In the Western literature a Noetherian operator is usually called a Fredholm operator. The index of such an operator is the number $ \chi ( A) = n ( A) - d ( A) $. The product of two Noetherian operators $ A $ and $ B $ is again a Noetherian operator, and $ \chi ( A B ) = \chi ( A) + \chi ( B) $. In the first concrete applications (see Noether's paper [1]) the index was calculated as a winding number associated with a certain continuous function. The computation of the index for different classes of operators is an important problem in modern mathematics (see e.g. [a1]).

References

[a1]	R.S. Palais, "Seminar on the Atiyah–Singer index theorem" , Princeton Univ. Press (1965)
[a2]	I.C. [I.Ts. Gokhberg] 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 (1957) pp. 43–118
[a3]	I. [I.Ts. Gokhberg] Gohberg, N. Krupnik, "Einführung in die Theorie der eindimensionalen singulären Integraloperatoren" , Birkhäuser (1979) (Translated from Russian)
[a4]	S. Goldberg, "Unbounded linear operators" , McGraw-Hill (1966)
[a5]	T. Kato, "Perturbation theory for nullity, deficiency and other quantities of linear operators" J. d'Anal. Math. , 6 (1958) pp. 261–322
[a6]	S.G. Krein, "Linear equations in Banach spaces" , Birkhäuser (1982) (Translated from Russian)