↑ arvau

Wikipedia / 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 be a linear operator with domain. Let id denote the identity operator on X, let

is said to be a regular value if the following three statements are true:

  1. is injective to its image has an inverse called the resolvent;

  2. is a bounded linear operator;

  3. is defined on a dense subspace of X 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

and subject to a mutually singular spectral decomposition into the point spectrum (when condition 1 fails), the continuous spectrum (when condition 2 fails) and the residual spectrum (when condition 3 fails).

is a closed operator be surjective.

Properties

  • 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.

Notes

References

  • (See section 8.3)

See also