The spectral theorem allows for a decomposition of the spectrum of a self-adjoint operator $T: D(T) \to H$, different form the standard one (continuous spectrum $\sigma_c(T)$, point spectrum $\sigma_p(T)$ and residual spectrum $\sigma_r(T)$ absent for self-adjoint operators), consisting in the discrete spectrum
$$ \sigma_d(T) := \left\{ \lambda \in \sigma(T) \:\: \left| dim\left(P^{(T)}_{(\lambda-\epsilon, \lambda +\epsilon)}(H)\right.\right)\:\:\mbox{is finite for some $\epsilon >0$} \: \:\right\}\:,$$
plus the essential spectrum
$\sigma_{ess}(T) := \sigma(T) \setminus\sigma_{d}(T)$.
It is not hard to see that $\lambda \in \sigma_d(T)$ $\;\Leftrightarrow\;$ $\lambda$ is an isolated point in $\sigma(T)$, and as such, $\lambda$ is an eigenvalue for $T$ with finite-dimensional eigenspace. One has $\sigma_d(T) \subset \sigma_p(T)$. In general, though, the opposite inclusion fails, for instance because there may be non-isolated points in $\sigma_p(T)$.
A third spectral decomposition for $T: D(T) \to H$ self-adjoint arises by splitting the Hilbert space into the closed span $H_{p}$ of the eigenvectors and its orthogonal complement: $H = H_{p}\oplus H_{p}^\perp$. Both $H_{p}\cap D(T)$ and $H_{p}^\perp \cap D(T)$ are easily $T$-invariant. With the obvious symbols:
$$T = T|_{H_{p}} \oplus T|_{H_{p}^\perp}\:.$$
One calls purely continuous spectrum the set $\sigma_{pc}(T):= \sigma(T|_{H_{p}^\perp})$, where for simplicity $T|_{H_p^\perp}$ stands for $T|_{D(T)\cap H_p^\perp}$ here and in the sequel.
Then $\sigma(T) = \overline{\sigma_p(T)} \cup \sigma_{pc}(T)$. Note that the latter is not necessarily a disjoint union, and in general $\sigma_{pc}(T)\neq \sigma_c(T)$.
The fourth spectral decomposition of $T: D(T) \to H$ on the Hilbert space $H$ (and even on a normed space), is that into approximate point spectrum
$$\sigma_{ap}(T) $$ $$ := \left\{ \lambda \in \sigma(T)\:\:|\:\:\mbox{$(T-\lambda I)^{-1}: Ran(T-\lambda I) \to D(T)$ does not exist or is not bounded}\right\}$$
and purely residual spectrum
$\sigma_{pr}(T) := \sigma(T)\setminus \sigma_{ap}(T)$.
The unboundedness of $(T-\lambda I)^{-1}$ is equivalent to the existence of $\delta>0$ with $||(T-\lambda I)\psi || \geq \delta ||\psi||$ for any $\psi \in D(T)$, so we immediately see how the next result comes about, thereby justifying the names: $\lambda \in \sigma_{ap}(T)$ $\;\Leftrightarrow\;$ there is a unit $\psi_\epsilon \in D(T)$ such that
$$||T\psi -\lambda \psi || \leq \epsilon\:$$
for any $\epsilon >0$. For self-adjoint operators the above holds for any $\lambda \in \sigma_c(T)$, but clearly also for $\lambda \in \sigma_p(T)$; since $\sigma(T) = \sigma_p(T) \cup\sigma_c(T)$ in this case, we conclude $\sigma_{ap}(T) =\sigma(T)$ and
$\sigma_{pr}(T) = \emptyset$ for every self-adjoint operator.
The last partial spectral classification for self-adjoint operators descends from Lebesgue's theorem on the decomposition of regular Borel measures on $\mathbb R$. If $T$ is self-adjoint on the Hilbert space $H$ and $\mu_\psi$ is the spectral measure of the vector $\psi$, we define the sets (all closed spaces):
$H_{ac}:=\{\psi \in H \:|\: \mu_\psi \mbox{ is absolutely continuous for Lebesgue's measure}\}$,
$H_{sing}:=\{\psi \in H \:|\: \mu_\psi \mbox{ is singular and continuous for Lebesgue's measure}\}$,
$H_{pa}:=\{\psi \in H \:|\: \mu_\psi \mbox{ is purely atomic}\}$.
Then we define $\sigma_{ac}(T) := \sigma(T|_{H_{ac}})$, $\sigma_{sing}(T) := \sigma(T|_{H_{sing}})$ respectively called absolutely continuous spectrum of $T$ and singular spectrum of $T$.It turns out that $\sigma_{ac}(T) \cup \sigma_{sing}(T) = \sigma_{pc}(T)$ and$\overline{\sigma_p(T)}= \sigma(T|_{H_{pa}})$.
Q&A (4871)
