Talk:Essential spectrum

message: 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)
replies: text: 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)
replies: text: Great! This is the most satisfactory solution. Oleg Alexandrov 17:53, 13 Jan 2005 (UTC)
replies: text: 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)
thread_title: More comments 

message: 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)
thread_title:  Move [[Essential spectrum#The_essential_spectrum_of_closed_operators_in_Banach_spaces]] to the definition

message: 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)
thread_title:  Hilbert space case