# Noetherian operator

A linear operator (with closed range) that is simultaneously -normal and -normal (see Normally-solvable operator). In other words, a Noetherian operator is a normally-solvable operator of finite -characteristic (, ). The index (cf. Index of an operator) of a Noetherian operator is also finite. The simplest example of a Noetherian operator is a linear operator acting from to . 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 has either no non-trivial solutions or a finite number of linearly independent solutions; and

2) the inhomogeneous equation is either solvable for any right-hand side , or for its solvability it is necessary and sufficient that , , where 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 is a Noetherian operator.

The property of being Noetherian is stable: If is a Noetherian operator and is a linear operator of sufficiently small norm or is completely continuous, then is also Noetherian, and .

Suppose that , where is the space of linear operators from to , is Noetherian. Then there is the direct decomposition

where is the null space of , is the range of and . The general solution of the equation , , is of the form , where , on (the restriction of ) and is arbitrary. If is Noetherian with -characteristic , then is Noetherian with -characteristic .

#### 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)