Perhaps this is a silly question to ask but it's been on my mind for a bit. When I took my first course in functional analysis a year ago, we covered spectral theory. Particularly, we covered spectral decomposition and discussed the point, continuous and residual spectrum cases. Fast forward to about a month ago in measure theory. We covered the Lebesgue-Radon-Nikodym theorem and showed that measures decompose into a discrete piece, an absolutely continuous piece and a singular (with respect to the discrete) continuous piece. The similarity in the decomposition and nomenclature seemed to suggest to me that maybe there were connections between them however I haven't seen anything suggesting this in my texts. Is this a coincidence or is there something underlying these two seemingly disjoint topics?

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    $\begingroup$ I think you might want to look into spectral measures... you'll see the measure theoretic topics you're mentioning come back in spectral theory $\endgroup$ – Tom Dec 1 '13 at 3:36
  • $\begingroup$ @Tom do you have a good resource for this? This is the first I've heard of spectral measures. $\endgroup$ – Cameron Williams Dec 1 '13 at 3:37
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    $\begingroup$ My favorite book which touches upon the topic is Volume I of "Methods of Modern Mathematical Physics" by Reed and Simon. $\endgroup$ – Tom Dec 1 '13 at 3:39
  • $\begingroup$ @Tom Great! I'll check it out after finals. Thanks for the resource. $\endgroup$ – Cameron Williams Dec 1 '13 at 3:42

The point/continuous/residual decomposition of spectrum applies to bounded operators on Banach spaces in general. The link from this more general case is not very direct.

For self-adjoint operators $T$ on a (separable) Hilbert space, however, it is immediate. Every such operator is unitarily equivalent to multiplication by $x$ on a countable Hilbert direct sum

$$ \oplus_i L^2(\mathbb{R}, d \mu_i), $$

where $\{ \mu_i \}_i$ is a family of Radon measures. The support of $\mu_i$'s gives classification of $\sigma(T)$. If a $\mu_i$ is absolutely continuous/discrete/singular w.r.t. the Lebesgue measure, its support belongs to the absolutely continuous/discrete/singular parts of $\sigma(T)$.

Unlike the Banach space case, this union is not disjoint, due to multiplicity being taken into account.


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