Harman Patil (Editor)

Resolvent set

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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.

Contents

Definitions

Let X be a Banach space and let L : D ( L ) → X be a linear operator with domain D ( L ) ⊆ X . Let id denote the identity operator on X. For any λ ∈ C , let

L λ = L − λ i d .

λ is said to be a regular value if R ( λ , L ) , the inverse operator to L λ

  1. exists, that is, L λ 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:

ρ ( L ) = { λ ∈ C | λ  is a regular value of  L } .

The spectrum is the complement of the resolvent set:

σ ( L ) = C ∖ ρ ( L ) .

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

  • The resolvent set ρ ( L ) ⊆ C of a bounded linear operator L is an open set.

    • References

    Resolvent set Wikipedia
    (Text) CC BY-SA