Talk:Decomposition of spectrum (functional analysis)

message: Some incorrect statements made in the article were changed, perhaps someone can elaborate on the details. 
:above unsigned comment was added by me. about two months(?) ago. Mct mht 22:54, 5 May 2006 (UTC)

message: I think there's an error in the proof that the spectrum of T_h equals the essential range of h. the f_n are not in L^p unless we find a subset of S_n of finite positive measure and take the characteristic funktion of this subset. we have to require that the measure \mu is locally finite, i.e. every measurable set of positive measure contains a finite measurable set of positive measure.
 
for an counterexample, consider S = \{x_1,x_2\}, where x_1 has finite and x_2 has infinite measure. then L^p(\mu) consits of those functions which vanish in x_2. let h be the characteristic function of \{x_2\}. then the essential range of h is \{0,1\}, but T_h=0 and hence the spectrum is  \{0\}. ---oo- 23:58, 14 February 2007 (UTC)
replies: text: yes, you're right. article should be modified. Mct mht 00:06, 15 February 2007 (UTC)
replies: text: well, i am gonna change it to σ-finite. Mct mht 08:23, 6 March 2007 (UTC)

text: Another error seems to be in the phrase "isolated point of the spectrum", say g\in C[0,1] (real-valued just for kicks), then by continuity, the image of g is the same as the essential image, and since [0,1] is compact and connected, the image is a compact, connected subset of \mathbb{R}, and an isolated point would be if the image were a single point, in which case, yes that point would be in \sigma_p(T_g), however, say that the support of g is not all of [0,1] and say g is not identically zero, examples of such functions are readily available. Then take f:[0,1]\to\mathbb{R} with disjoint support, which is possible because the support of g is a compact subset of [0,1], and hence a disjoint union of closed intervals (finite), then gf=0 contradicts T_g 1-1, so 0 is in the point spectrum even though g may not be constant, i.e. a point spectrum value may not be an isolated point. As a result, I'm going to remove the language about being isolated and just leave the part that explains what it really means.  — Preceding unsigned comment added by Guardian of Light (talk • contribs) 16:23, 2 November 2011 (UTC)
thread_title:  spectrum of multiplication operator 

message: the article discrete spectrum could be merged with this one. mathematically, continuous spectrum can mean several things (similar but possibly distinct, depending on the particular operator). maybe the article can be expanded to include couple typical break-ups of the spectrum that one might encounter in the literature. Mct mht 12:15, 14 August 2006 (UTC)
replies: text: i forked those remarks to continuous spectrum. Mct mht 08:08, 15 August 2006 (UTC)
replies: text: Thanks very much for your work. The various kinds of spectrum have always confused me, and perhaps next time I need to look into this, I can use this page for some guidance (hint: please add references!). I wonder though why you used the title Decomposition of spectrum (functional analysis) instead of Decomposition of spectrum. Do you expect other pages with the title "Decomposition of spectrum (something)"? Otherwise, disambiguation is not necessary. -- Jitse Niesen (talk) 12:38, 15 August 2006 (UTC)
replies: text: no particular reason why i used that title, maybe you're right. i just thought since physicists speak of spectra, in non-rigorous fashion, all the time, better to specify. some references are Dunford & Schwartz, and Reed & Simon, will add them now. Mct mht 12:54, 15 August 2006 (UTC)
thread_title:  suggest merge and expand 

message: Why was this redirected to from "Continuous Spectrum" if they're separate articles? —The preceding unsigned comment was added by 199.197.111.212 (talk) 17:01, 24 April 2007 (UTC)
replies: text: sry, i moved the article but not the talk page. will be fixed shortly. Mct mht 08:24, 2 May 2007 (UTC)
thread_title:  Redirect 

message: I have some issues relating to the discussion of self-adjoint operators on Hilbert spaces.
Usually, one means by continuous spectrum, the union of the absolutely continuous spectrum
and the singular continuous spectrum. On the same note by singular spectrum the union
of the pure point spectrum and the singular continuous spectrum. So the notation with
H_{cont} for the subspace of the AC spectrum is a little bit confusing. The same is
true for the notation H_{sing} for the singular continuous spectrum.
Furthermore, the discrete spectrum corresponds to what physicist call bound states.
The pure point spectrum corresponds to "localization" in a vague sense. In fact
one means by localization also that the eigenfunctions are exponentially decaying,
or sometimes dynamical properties.
In order to have a relation to the physics it might be nice to have some reference
to the RAGE theorem! This theorem tells us how the Schrödinger flow
evolves depending on the spectral type of the operator.
And the claim that L^2(\mathbb{R}) functions decay is just plain
wrong. E.g. the function
 f(x) = \begin{cases} n & x \in [n, n+\frac{1}{n^4}] \\ 0 & else \end{cases} 
is a counter example.
Can I just go ahead and edit this article?
ElMaison (talk) 04:39, 30 March 2008 (UTC)
replies: text: Go for it. Or as we say here, Be Bold! People will complain if they do not like what you're doing, but if nobody edits the article, it will never improve. And obviously, if you change e.g. H_{cont} to H_{ac}, try to keep the whole article consistent. -- Jitse Niesen (talk) 10:16, 1 April 2008 (UTC)

text: The example of a non-decreasing L^2 function occurs in a confusing place and lacking context. When reading the article I got the impression that the function is supposed to be an example of an eigenfunction corresponding to a point in the point spectrum, but which is not localized. I had to go to the discussion to understand that it was really trying to show that L^2 does not imply decreasing. Actually it's not really successful on that count either, because each function in the sequence is compactly supported. Moreover, you would not normally call those functions 'increasing' because they are not monotone...  — Preceding unsigned comment added by 81.224.188.19 (talk) 00:38, 14 November 2021 (UTC)
thread_title:  Some Issues 

message: I don't understand what is going on with the weird definition of the spectrum here. It is stated that T-lambda is invertible if it is bounded below and has dense range. But if T-lambda is bounded below, then it has closed range (consider a cauchy sequence in the range, the corresponding preimages will also be a cauchy sequence). So, the definition is just that T-lambda is bounded below and is onto, which thanks to the closed graph theorem just says that T-lambda is bijective. It is also said that bijectivity is stronger than being invertible, which is not true.
--Roggles (talk) 13:43, 24 November 2008 (UTC)
thread_title:  "Incorrect" discussion of invertibility? 

message: It is claimed that if T^* is the adjoint of T then sigma(T^*)=sigma(T).  I have only seen sigma(T^*) subset sigma(T), not equality; either a proof should be supplied or a citation.  —Preceding unsigned comment added by 193.188.46.249 (talk) 08:47, 24 March 2009 (UTC)
thread_title:  Spectrum of Adjoint Operator 

message: The article should focus on the topic (*decomposition* of the spectrum) for readers who are likely to come here; namely people who already know the definition of spectrum of a functional operator.  There is a separate article for that definition; this article should link to that, keeping at most a discrete one-line refresher sentence. --Jorge Stolfi (talk) 18:03, 5 February 2013 (UTC)
thread_title:  Definition of spectrum should not be here