Different terminology for spectrum?

Question

The common terminology we get used to on spectrum of linear operators is

• point spectrum $ \sigma_p $,
• continuous spectrum $ \sigma_c $,
• residual spectrum $ \sigma_r $ .

However, we recently notice an terminology of another type:

• purely point spectrum $ \sigma_{pp} $ ,
• absolutely continuous spectrum $ \sigma_{ac} $ ,
• singularly continuous spectrum $ \sigma_{sc} $ .

The latter type is defined with respect to decomposition measure and orthogonal decomposition of spectral projections. For details, see for instance Chapter 3.3 in

"G. Teschl: Mathematical Methods in Quantum Mechanics - With Applications to Schrödinger Operators (2014)".

Question: What is the relation between above two terminologies, can we give an exact correspondence for them?

Edit: The relation I have found so far is $ \sigma_{pp} = \overline{\sigma_{p}} $.

One possibly useful result: Chapter IV, Proposition 1.16 in

[Engel, Nagel: One-Parameter Semigroups for Linear Evolution Equations (2000)]

However, I can not see clearly how to use this result to explain the connections between the two terminologies.

Answer 1

I managed to read up the definitions in the book you described (link). The two kinds of spectra really encode different things, and I do not believe they should be compared.

The first remark I would make is that the spectra of the "second kind" only make sense for self-adjoint (or normal) operators, that is operators where you have a spectral measure, whereas the spectra of the "first kind" can always be defined. Note however that if $A$ is self-adjoint that then $\sigma_r(A)=\emptyset$ always. That is for spectra of the "first kind" we only have access to $\sigma_p$ and $\sigma_c$ when we want to compare with $\sigma_{pp},\sigma_{ac},\sigma_{sc}$.

The second remark is that by definition $\sigma_p, \sigma_{c},\sigma_r$ are always disjoint, while $\sigma_{pp},\sigma_{ac},\sigma_{sc}$ may all overlap, indeed situations are possible where $\sigma_{pp}(A)=\sigma_{ac}(A)=\sigma_{sc}(A)$ (although not in separable Hilbert spaces, excepting the $0$ Hilbert space).

With these out of the way there are some things that can be said:

1. $\overline{\sigma_p(A)} =\sigma_{pp}(A)$.
2. If $H$ is separable and $A$ self-adjoint, then $\overline{\sigma_{c}(A)}\supseteq\sigma_{ac}(A)\cup \sigma_{sc}(A)$, although there are no restrictions on what can be in $\sigma_{ac}$ vs $\sigma_{sc}$ (both can be empty, or any one of the two can be empty, but I do not believe they can be the same in a seperable Hilbert space).
3. If $H$ is not separable you have no relation at all. For example $\sigma_c(A)$ can be empty but $\sigma_{ac}(A)$ and $\sigma_{sc}(A)$ can be anything.

If you are interested in these comments then I can describe how you can see them and the examples realising them in more detail. Lets describe a very singular example first:

Let $$H=\ell^2([0,1])\oplus L^2([0,1],dx) \oplus \bigoplus_{\alpha\in [0,1]}L^2([0,1], d\mu_\alpha)$$ where $\ell^2([0,1])$ is the space of functions $f:[0,1]\to\Bbb C$ so that $\sum_{x\in[0,1]}|f(x)|^2<\infty$ (this is a very big space), $L^2([0,1],dx)$ is the usual $L^2$ space and $L^2([0,1],d\mu_\alpha)$ the $L^2$ of a singular measure $d\mu_\alpha$ on $[0,1]$ for which $\alpha$ is in the support. Now define the bounded self-adjoint operator $M_x:H\to H$, given by the formula $M_x(f) = ( x\mapsto x\cdot f(x))$ (note that this formula makes sense for each summand of $H$).

You can check that $\sigma_p(M_x)=[0,1]=\sigma(M_x)$ and $\sigma_c(M_x)=\sigma_r(M_x)=\emptyset$, but also $\sigma_{pp}(M_x)=\sigma_{ac}(M_x)=\sigma_{sc}(M_x)=[0,1]$. If you remove any of the summands in the Hilbert space, you will make the corresponding $\sigma_{pp}$, $\sigma_{ac}$, $\sigma_{sc}$ empty.