Resolvent set

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.

Definitions

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

is said to be a regular value if , the inverse operator to

  1. exists, that is, is injective;
  2. is a bounded linear operator;
  3. is defined on a dense subspace of X.

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).

Properties

References

External links

See also

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