# Normally-solvable operator

A linear operator with closed range. Let be a linear operator with dense domain in a Banach space and with range in a Banach space . Then is normally solvable if , that is, if is a closed subspace of . Let be the adjoint of . For to be normally solvable it is necessary and sufficient that , that is, that the range of is the orthogonal complement to the null space of .

Suppose that

(*) |

is an equation with a normally-solvable operator (a normally-solvable equation). If , that is, if the homogeneous adjoint equation has only the trivial solution, then . But if , then for (*) to be solvable it is necessary and sufficient that for all solutions of the equation .

From now on suppose that is closed. A normally-solvable operator is called -normal if its null space is finite dimensional . A normally-solvable operator is called -normal if its deficiency subspace is finite dimensional . Operators that are either -normal or -normal are sometimes called semi-Fredholm operators. For an operator to be -normal it is necessary and sufficient that the pre-image of every compact set in is locally compact.

Suppose that is compactly imbedded in a Banach space . For to be -normal it is necessary and sufficient that there is an a priori estimate

It turns out that an operator is -normal if and only if is -normal. Then . Consequently, if is compactly imbedded in a Banach space , then is -normal if and only if there is an a priori estimate

The pair of numbers is called the -characteristic of . If a normally-solvable operator is -normal or -normal, the number

is called the index of the operator . The properties of being -normal and -normal are stable: If is -normal (or -normal) and is a linear operator of small norm or completely continuous, then is also -normal (respectively, -normal).

#### References

[1] | F. Hausdorff, "Grundzüge der Mengenlehre" , Leipzig (1914) (Reprinted (incomplete) English translation: Set theory, Chelsea (1978)) |

[2] | F. Atkinson, "Normal solvability of equations in Banach space" Mat. Sb. , 28 : 1 (1951) pp. 3–14 (In Russian) |

[3] | S.G. Krein, "Linear differential equations in Banach space" , Transl. Math. Monogr. , 29 , Amer. Math. Soc. (1971) (Translated from Russian) |

#### Comments

#### References

[a1] | 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 |

[a2] | S. Goldberg, "Unbounded linear operators" , McGraw-Hill (1966) |

[a3] | T. Kato, "Perturbation theory for nullity, deficiency and other quantities of linear operators" J. d'Anal. Math. , 6 (1958) pp. 261–322 |

[a4] | S.G. Krein, "Linear equations in Banach spaces" , Birkhäuser (1982) (Translated from Russian) |

**How to Cite This Entry:**

Normally-solvable operator. V.A. Trenogin (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Normally-solvable_operator&oldid=15607