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Concept
Resolvent set
Summary
In linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important role in the resolvent formalism.
Let X be a Banach space and let be a linear operator with domain . Let id denote the identity operator on X. For any , let
A complex number is said to be a regular value if the following three statements are true:
is injective, that is, the corestriction of to its image has an inverse ;
is a bounded linear operator;
is defined on a dense subspace of X, that is, has dense range.
The resolvent set of L is the set of all regular values of L:
The spectrum is the complement of the resolvent set:
The spectrum can be further decomposed into the point/discrete spectrum (where condition 1 fails), the continuous spectrum (where conditions 1 and 3 hold but condition 2 fails) and the residual/compression spectrum (where condition 1 holds but condition 3 fails).
If is a closed operator, then so is each , and condition 3 may be replaced by requiring that is surjective.
The resolvent set of a bounded linear operator L is an open set.
More generally, the resolvent set of a densely defined closed unbounded operator is an open set.
Official source
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