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--- hamburgminicourse2017:the_spectrum_of_operators [2017/04/03 16:13] – markus
+++ hamburgminicourse2017:the_spectrum_of_operators [2017/04/04 17:38] (current) – admin
@@ Line 33: / Line 33: @@
   - Finally the last case might hold with $A-\lambda\mathrm{id}$ having a non-dense range, so the $\varepsilon_i$ are limited to certain subspaces and $\lambda$ is called part of the //residual spectrum//. This is possible for bounded operators but not for self-adjoint ones.
-**Note.** The [[wp>Lebesgue decomposition theorem]] allows a different [[wp>Decomposition_of_spectrum_(functional_analysis)#Decomposing_the_spectrum|partitioning of the spectrum]] into an //absolutely continuous//, //singular continuous//, and //pure point// part.
+**Note.** The spectral theorem and the [[wp>Lebesgue decomposition theorem]] allow a different [[wp>Decomposition_of_spectrum_(functional_analysis)#Decomposing_the_spectrum|partitioning of the spectrum]] for normal operators into an //absolutely continuous//, //singular continuous//, and //pure point// part.
 For a self-adjoint operator it holds that all $z \in \mathbb{C}$ with $\Im z \neq 0$ are in the resolvent set and the bounded operator $(A-z\mathrm{id})^{-1}$ is called a //[[wp>Resolvent formalism|resolvent]]// that establishes a very useful link to compex analysis. The norm of this operator obeys the following estimate: