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Definition of the residual spectrum

  1. Mar 21, 2012 #1
    If [itex]A[/itex] is a bounded operator on a Hilbert space [itex]H[/itex], isn't the following true of the residual spectrum [itex]\sigma_r(A)[/itex]:

    [itex]\lambda \in \sigma_r(A)[/itex] iff [itex](\forall \psi \in H, \psi \neq 0)((\lambda - A) \psi \neq 0)[/itex] iff [itex]\ker (\lambda - A) = \{0\}[/itex] iff [itex]\lambda - A[/itex] is injective?

    So isn't the condition that [itex]\lambda[/itex] isn't an eigenvalue equivalent to the condition [itex]\lambda - A[/itex] is injective?
  2. jcsd
  3. Mar 21, 2012 #2
    No, [itex]\lambda[/itex] is in the residual spectrum if [itex]\lambda-A[/itex] is injective AND does not have dense range. You're missing here that [itex]\lambda-A[/itex] can not have dense range.

    This is true.

    It seems you're mistaking the residual spectrum for the part of the spectrum which are not eigenvalues. This is not true. The spectrum can be divided in 3 parts:

    - The point spectrum: all the eigenvalues
    - The continuous spectrum: These are the [itex]\lambda[/itex] such that [itex]\lambda - A[/itex] is injective and has dense range.
    - The residual spectrum:These are the [itex]\lambda[/itex] such that [itex]\lambda - A[/itex] is injective and does not have dense range.

    So the elements which are not eigenvalues are either contained in the continuous or the residual spectrum.
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