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What is the precise relation between continuous spectrum, singular spectrum, etc.?

+ 4 like - 0 dislike
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In my answer to this question, I confused the residual spectrum and the singular spectrum, because I had identified the classification of the spectrum of an operator into point spectrum, continuous spectrum, and 
residual spectrum with the  classification of the spectrum of a diagonalizable operator into absolutely continuous spectrum, pure point spectrum, and singular spectrum.

What is the precise relation between these concepts?

asked Mar 3, 2015 in Mathematics by Arnold Neumaier (12,355 points) [ no revision ]

2 Answers

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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$\overline{\sigma_p(T)}= \sigma(T|_{H_{pa}})$.

answered Mar 4, 2015 by Valter Moretti (2,025 points) [ revision history ]
edited Mar 4, 2015 by Valter Moretti

Thanks. Please polish the latex a bit (you just did it in part; the \index stuff is still there). I'd also like to see the coarsest partition of which all the various parts are unions of, and an indication of (or reference to) an example for a member of each part.

Sorry, I am too busy for adding further details now. These decompositions appear in Reed-Simon vol I and I guess vol III.

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As I recall it, my source being the book by Reed and Simon, the distinction between point, continuous and residual spectrum is possible for any linear operator $T$ in a Banach space and it is related to the different motivations for which a certain complex value $\lambda$ does not belong to the resolvent set $\rho(T)$ (as already discussed in the other question).

The distinction between pure point, absolutely continuous and singularly continuous spectrum is instead meaningful only for normal operators on Hilbert spaces, because it is related to the spectral theorem. In particular, the spectrum is characterized by the behavior of the spectral measures associated to the operator. The spectral measures are measures on $\mathbb{R}$, and any measure $\mu$ on $\mathbb{R}$ can be decomposed in three mutually singular parts: one pure point $\mu_{pp}$ (i.e. supported on isolated points of the real axis), one absolutely continuous w.r.t. the Lebesgue measure $\mu_{ac}$ (it means that there is a function $f$ locally integrable such that $d\mu_{ac}=fdx$ where $dx$ is the Lebesgue measure) and one that is singular continuous w.r.t. the Lebesgue measure $\mu_{sc}$ (its support has no isolated points and there exist a set $S$ such that $Leb(\mathbb{R}\setminus S)=0$ and $\mu_{sc}(S)=0$, $Leb$ being the Lebesgue measure). The spectrum of a normal operator $A$ can therefore be decomposed into $\sigma(A)=\sigma_{pp}(A)+\sigma_{ac}(A)+\sigma_{sc}(A)$, where each part is the support of the corresponding part of the spectral measures.

Another decomposition of the spectrum is very often utilized for self-adjoint/normal operators, and it is related to spectral projections rather than spectral measures. It is the notion of discrete spectrum and essential spectrum. Without giving too much details, the definition of Reed and Simon corresponds to the following: a value is in the discrete spectrum if it is an eigenvalue of finite multiplicity and it is an isolated point of the spectrum; it is in the essential spectrum otherwise.

answered Mar 4, 2015 by yuggib (360 points) [ no revision ]
reshown Mar 4, 2015 by yuggib

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