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 .
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).
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Normally-solvable operator. V.A. Trenogin (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Normally-solvable_operator&oldid=15607