Talk:Essential spectrum

Definition
Are you sure about the definition you wrote? I have a reference (Sandstede, Stability of travelling waves) which states that the essential spectrum of the operator T is the spectrum minus the point spectrum, and the point spectrum is where the operator $T - \lambda I$ is Fredholm with index zero. It is of course well possible that different definitions float through the literature. -- Jitse Niesen 21:05, 12 Jan 2005 (UTC)

Very good point. First, I must say that I got that info from the internet. I had checked a couple of papers and thought that was enough. Now, the internet is not always reliable, so when I got your message I went and did more searches on google with essential spectrum fredholm google.

One reference, at says:
 * The essential specturm of D is the set of lambda, such that D-\lambda I is not semi-Fredholm where semi-Fredholm means by definition here that the kernel is finite dimensional and the range is closed (there is no ambiguity because the operator is L^2 is self-adjoint).

So they use semi-Fredholm there.

The other papers I found, all use the same definition as I wrote, that is, with lambda-D not being Fredholm, where D is the operator.
 * http://www.imar.ro/~golenia/Fichiers/dpo.pdf (page 2, top)
 * http://sunsite.wits.ac.za/mmoller/ps/mn202-94.ps (page 2, bottom)
 * http://parallel.hpc.unsw.edu.au/rks/docs/specbolt/node2.html (this is html, they define the spectrum and cite a paper for definition)
 * http://math.albany.edu:8000/~kzhu/spectrum.pdf (page 1, bottom)
 * http://www.math.snu.ac.kr/~wylee/HaL1.pdf (page 1, top)

This still does not prove that the def I gave is correct, and the first definition is actually slightly different. The definition in your book could be right, but there is something I don't like in it. From it, it follows that the spectrum is just the point spectrum and the essential spectrum. From what I know, there is a third kind of spectrum, called continous spectrum, which is neither of the two. I will think more about this. Oleg Alexandrov 23:52, 12 Jan 2005 (UTC)

For an operator on a Hilbert space, the essential spectrum is the spectrum modulo the compacts, i.e. the spectrum in the Calkin algebra. That is


 * $$ \lambda \in \operatorname{ess}\,\sigma(T) \iff \exists K \in \mathcal{K}(H), T+K - \lambda \mbox{ is invertible }.$$


 * The previous condition is not equivalent to being invertible modulo the compacts. (A counterexample is the unilateral shift which is Fredhlom of index 1). The correct characterization of essential spectrum is


 * $$ \lambda \in \operatorname{ess}\,\sigma(T) \iff \not \exists T' \in \mathcal{L}(H), (T-\lambda I ) T' = T' (T - \lambda I) = I \pmod{\mathcal{K}(H)}.$$


 * This says exactly that T - &lambda; is not invertible modulo the compacts, which is equivalent to T - &lambda; being not Fredholm. But the rest of what I said is true (I hope). I also hope I haven't confused everybody...CSTAR 04:56, 13 Jan 2005 (UTC)

For self-adjoint operators this boils down to the disjunctive condition (disjunctive = OR)


 * &lambda; is a limit point of the spectrum
 * &lambda; is an eigenvalue of infinite multiplicity (could be isolated).

CSTAR 00:08, 13 Jan 2005 (UTC)

But this is not true for general operators, as claimed in the section The Discrete Spectrum. For example, there are compact (quasi-nilpotent) operators whose spectrum is 0, hence isolated, yet that value is not an eigenvalue. — Preceding unsigned comment added by Chrystomath (talk • contribs) 07:56, 2 June 2012 (UTC)

Now, it would be interesting to know how this compares to the definition I wrote on essential spectrum. Oleg Alexandrov 00:11, 13 Jan 2005 (UTC)


 * Fredholm operators are those which are invertible modulo the compacts so your definition is right, i.e. it extends to Banach spaces the definition above for operators on Hilbert spaces. Notice that in general Banach spaces, the ideal of compact operators is not the same as the norm closure of the ideal of operators of finite rank.  Unfortunately I'm away from my library, so I can't give you hard references.CSTAR 00:17, 13 Jan 2005 (UTC)


 * Oops and I made a mistake, the condition above should be is not invertible.CSTAR 00:19, 13 Jan 2005 (UTC)


 * I probably got the quantifier wrong too. But in any case, essential spectrum corresponds exactly to failure of invertibility modulo the compacts and your definition (Oleg) is correct At this hour of the day, my ability to manipulate quantifiers has decreased to nil.CSTAR 00:29, 13 Jan 2005 (UTC)

Colleagues, I understand that it has been at least a year since any of you made contributions to this page. Overall, I like it, however, the exposition is not quite as clear as I prefer. For example, it is stated that the spectrum of T consists of those values $$ \lambda \in \mathbb{C} $$ such that $$ T-\lambda $$ maps a norm 1 sequence onto a sequence which converges to zero. While this is true for the self adjoint operators being considered (and more generally for normal operators), I think it is a faux pas to fail to state that explicitly when making that claim, even though $$ T=T^* $$ may have been previously stated. Loreaujy (talk) 13:58, 3 August 2010 (UTC)

More comments
Now for a self-adjoint operator A if it is Fredholm, it has index 0. So the condition that requires index to be 0 is superfluous in this case. It is possible that as Jitse Niesen suspects, there may be more than one definition of essential spectrum. I've only worked with essential spectrum for s.a. operators (for instance Hamiltonians). So we're basically back where we started. And I'm away from a library, so I can't really consult anything.CSTAR 05:48, 13 Jan 2005 (UTC)


 * What CSTAR says matches with the little that I know from Reed and Simon, Functional Analysis I, but indeed they deal only with self-adjoint operators in that section. The Sandstede reference is about non-self-adjoint operators, but I'd like to be a bit more certain before positing this as the true and only definition. For the moment, I added self-adjoint to the article to make sure that we tell no untruths, okay? But I'd like to learn this once and for all, because all these different kinds of spectra confuse me mightily, so I might hit the library later. Oleg, the Reed & Simon reference gives two decompositions of the spectrum, namely point spectrum + absolutely continous spectrum + residual spectrum and essential spectrum + discrete spectrum, so that is where the residual spectrum fits in. Jitse Niesen 11:35, 13 Jan 2005 (UTC)


 * Great! This is the most satisfactory solution. Oleg Alexandrov 17:53, 13 Jan 2005 (UTC)


 * And one day, very far away, somebody knowing functional analysis should go and do work on the Spectrum of an operator page. Too hard proofs in there, and too little stuff about the "Classification of points in the spectrum". Oleg Alexandrov 17:59, 13 Jan 2005 (UTC)

Move Essential spectrum to the definition
This section provides several definitions of the essential spectrum, which are not all compatible with the definition above. I suggest this section replace the definition. Any content specific to the definition given could be moved to a new section specific to the properties of that definition (which appears to be the definition of $$\sigma_{\text{ess}, 3}$$ if I am not mistaken). These five definitions seem to be floating around on several pages (here, Spectrum (functional analysis), Decomposition of spectrum (functional analysis), etc.) though none of them have citations. Is this a standard notation? Does anyone have citations for any or all of these definitions? Shawsa7 (talk) 21:44, 28 July 2022 (UTC)

Hilbert space case
For the Hilbert space case, it's not necessary to restrict the discussion to self-adjoint operators. Exactly the same definition goes for bounded operators in general. Although some statements in the current version are particular to the self-adjoint case:


 * Weyl's definition of the spectrum: λ lies in the spectrum if and only if T - λ is not bounded below. Non-self adjoint operators can be bounded below but not have dense range, therefore not invertible. Take the unilateral shift, for example.


 * The "discrete spectrum", as defined in this article, would contain more than isolated eigenvalues of finite multiplicity for general bounded operators. (Maybe this definition only concerns the self-adjoint case.) E.g., take again the unilateral shift, which has closed unit disk as the spectrum, the circle as the essential spectrum, and no eigenvalues. Mct mht (talk) 15:34, 5 September 2012 (UTC)