Talk:Spectrum (functional analysis)

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Contents 1 Bold 2 TeX vs. Non-Tex markup 3 Spectrum categories are inconsistent 4 Operators on more general topological vector spaces 5 Spectrum of an operator

[edit] Bold

This page needs to be rewritten as it only discusses the spectrum of bdded operators. OoberMick 15:14, 18 Nov 2004 (UTC)

Huh? linas 6 July 2005 01:01 (UTC)

When I wrote that comment the remark about unbounded operators wasn't there, but still the article has a bias towards bounded operators. In fact the definition of the spectrum states "Let B be a complex Banach algebra containing a unit e". The definition of the spectrum does not require that B is a Banach algebra only a Banach space and such a definition, where B was a Banach Algebra would be equivelent to a definition on the Banach space B(X,X) of all bounded linear operators on a banach space X. OoberMick 11:09, 29 July 2005 (UTC)

Interesting. What is the definition of the spectrum of an element of a Banach space? Lupin 13:03, 29 July 2005 (UTC)

Appologies, what I've written isn't quite right. I should have wrote ...does not require that B is a Banach algebra only a space of operators on a Banach space and such a definition... But I wasn't completely wrong the dual space (space of linear operators) of a Banach space is indeed a Banach space :). With this is mind we can define the spectrum of a closed operator T in our Banach space X. (If T isn't closed, take it's closure... if T isn't closable the whole complex plane is the spectrum). We define the spectrum as the complementary set of the resolvent set, where the resolvent set is the set of complex numbers z such that (T − z) ^{− 1} exists and lives in B(X,X). So a complex number is an element of the spectrum if T − z is not invertable or (T − z) ^{− 1} is not densely defined or (T − z) ^{− 1} is not bounded.

So are you advocating a definition for any complex unital algebra, not necessarily normed or complete? That sounds feasible, but I don't know what you can say in that generality. Maybe the algebraists already have it in an article somewhere. Lupin 16:15, 1 August 2005 (UTC)

What I'm looking for is a defintion of the spectrum which is general enough to allow discussion of the spectrum of differential operators such as the Laplacian and the Schrödinger equation. My personal preference would be to define the sepctrum in terms of operators (as opposed to an element of an algebra) since this would be more consistent with the definition in finite dim where we discuss matrices on a vector space. It would then be possible to show that if we consider only bounded operators then this is a Banach algebra. OoberMick 08:59, 2 August 2005 (UTC)

I agree with OoberMick that the definition is lacking a bit and can be generalized. Below is the definition I use, from notes taken from Kreysig.

Let X be a non-trivial complex, normed linear space (not necessarily complete), and T a linear operator (not necessarily bounded) that maps from a domain in X into X. For a complex number λ, define T_λ = T − λI and the resolvent R_λ = (T_λ) ^{− 1}. From now on, I'll write R to mean R_λ.

Then the regular valies of the operator are complex numbers such that (1) R exists, (2) R is bounded, and (3) R is defined on a dense set in X. Non-regular values are the spectrum, which are divided into the point spectrum [where (1) doesn't hold], the continuous spectrum [where (1) and (3) hold but (2) doesn't], and the residual spectrum [where (1) holds but (3) doesn't hold]. Lavaka 18:16, 7 November 2006 (UTC)

[edit] TeX vs. Non-Tex markup

Re recent revert by User:Jitse Niesen of edits by User:MathKnight, titled :please see discussion on Wikipedia_talk:How to write a Wikipedia article on Mathematics." I actually liked MathKnight's tex-ification. -- linas 6 July 2005 01:01 (UTC)

The God given inviolable rule is that there shall no be any worship to PNG idols unless on a separate line. The scripture at Wikipedia:How to write a Wikipedia article on Mathematics confirms that. Ask God for MathML to be implemented in all browsers soon, that will solve the issue. Oleg Alexandrov 6 July 2005 03:24 (UTC)

[edit] Spectrum categories are inconsistent

The names of the three categories of the spectrum are listed here as point spectrum, approximate point spectrum, and compression spectrum. But these categories have other names. I seem to recall that the point spectrum is sometimes called the eigenvalue spectrum (I am unsure, but I seem to recall this is the case). The approximate point spectrum is also called the continuous spectrum, and the compression spectrum is also called the essential spectrum or the residual spectrum. Given that the names used in the article aren't consistently followed. Further on, the wikipedia links to the spectrum categories are named differently. My recommendation is that we use the names "point spectrum", "continuous spectrum" (though this seems defined somewhat differently than the article specific to continuous spectrum), and "essential spectrum" to match the links to the relevant wikipedia articles, and mention in the brief description under each section in this article, the alternate names for these sets. -- KarlHallowell 04:54, 1 November 2005 (UTC)

Ok, I'm somewhat incorrect above. The essential spectrum is actually the combined approximate point spectrum and the compression spectrum. There may be other errors as well. I'll look this material up and see if I can at least find all the commonly used terms. -- KarlHallowell 17:23, 3 November 2005 (UTC)

all those types of spectra names above, and their definitions, can be encountered in the literature. they are not necessarily the same, if one accepts the standard definitions. for example, let's say a is in the approx. pt. spectrum if T - a is not bounded below. On the other hand, a is in the continuous spectrum if T - a is injective and have dense range. So the continuous spectrum would be contained in the approx. pt. spectrum but the converse need not be true in general. the continuous spectrum is by definition disjoint from the residual spectrum while parts of the approx. pt. spectrum may lie in the residual spectrum.

for shifts on l ^p, the approx. pt. and continuous spectra coincide, it's the unit circle. Mct mht 09:17, 15 August 2006 (UTC)

[edit] Operators on more general topological vector spaces

What about operators on more general topological vector spaces (that are not Banach spaces)? Certainly the requirement that T - lambda is not invertible makes sense in that context. Crust 20:23, 15 November 2005 (UTC)

Seems like you get a big mess from the topology. For example, an operator might be invertible in the TVS (topological vector space), but not in the completion of the space. Maybe the space of all differentiable functions would make a good example? -- KarlHallowell 01:16, 16 November 2005 (UTC)

KarlHallowell, sounds like a good point. (Just to make sure I understand you correctly since it's been some time since I did functional analysis: I think what you are considering a situtation with U,V TVS's with U \subset V, U has the induced topology and U dense in V. Then your concern is that a continuous linear operator T: V->V might have a different spectrum than its restriction T:U->U.) Yes, I was thinking primarily of locally convex TVS's given by a family seminorms, such as various spaces of differentiable functions. I think my concern is similar to OoberMick's above. Crust 20:25, 16 November 2005 (UTC)

[edit] Spectrum of an operator

If we have a hermitian operator H that can be decomposed as

with H₁, H₂ hermitian, U₁, U₂ unitary, is there a way of combining the Ds together to give the original spectrum of H? --HappyCamper 03:49, 18 September 2006 (UTC)

in general, it sounds too much to hope for to me. if H_1 and H_2 commute, then the diagonalization of H is just D_1 + D_2. also, if we're talking about matrices, then there are perturbation-theoretic results from matrix analysis that give you some idea where the eigenvalues of H are located, in terms of eigenvalues of H_1 and H_2. Mct mht 05:40, 18 September 2006 (UTC)

It does sound like wishful thinking. But going along the lines of your latter point, do you have good resources for that? Ideally, something with error bounds would be nice. I'd like to check up on that. --HappyCamper 15:21, 18 September 2006 (UTC)

surely for matrices some estimates are possible. you might wanna look into books on matrix analysis. Horn and Johnson, both volumes, would certainly contain that kinda stuff. also, first chapters(chap. 1-2?) from Matrix Analysis by Bhatia (it's one of the GTM, i believe). Mct mht 18:36, 18 September 2006 (UTC)