Connections between Lebesgue-Radon-Nikodym decomposition and spectral decomposition

Question

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?

Answer 1

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.