Definition of the residual spectrum

In summary, the residual spectrum \sigma_r(A) can be defined as the set of \lambda such that \lambda - A is injective and does not have dense range. This condition is equivalent to the condition that \lambda is not an eigenvalue of A. The spectrum of a bounded operator A can be divided into three parts: the point spectrum, the continuous spectrum, and the residual spectrum. The residual spectrum contains elements that are not eigenvalues and do not have dense range.
  • #1
AxiomOfChoice
533
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?
 
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  • #2
AxiomOfChoice said:
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?

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.

So isn't the condition that [itex]\lambda[/itex] isn't an eigenvalue equivalent to the condition [itex]\lambda - A[/itex] is injective?

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.
 

1. What is the definition of the residual spectrum?

The residual spectrum is the collection of frequencies that remain after a signal has been filtered or processed in some way. It represents the components of the original signal that were not affected by the filtering or processing.

2. How is the residual spectrum different from the original signal?

The residual spectrum is different from the original signal because it only contains the frequencies that were not affected by the filtering or processing. This means that the amplitude and phase of the residual spectrum may be different from the original signal.

3. What is the importance of the residual spectrum in signal processing?

The residual spectrum is important in signal processing because it can provide insights into the characteristics of the original signal. It can also be used to evaluate the effectiveness of a filter or processing technique.

4. Can the residual spectrum be used to reconstruct the original signal?

No, the residual spectrum cannot be used to reconstruct the original signal. It only contains the frequencies that were not affected by the filtering or processing, so information about the amplitude and phase of the original signal is lost.

5. How is the residual spectrum calculated?

The residual spectrum is typically calculated by subtracting the filtered or processed signal from the original signal. This leaves behind the frequencies that were not affected by the filtering or processing, resulting in the residual spectrum.

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