Talk:Unbounded operator

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Is it really necessary to restrict the article to unbounded operators on a Hilbert space? I understand that's the most important case in mathematical physics, but it seems like a bias. -- Taku (talk) 10:57, 18 April 2009 (UTC)

As is written, "Some generalizations to Banach spaces and more general topological vector spaces are possible". Namely, "operator", "densely defined" and "closed" can be generalized immediately; however, "symmetric" and "self-adjoint" (the most important!) are for Hilbert spaces only (as far as I know).
For my opinion, you may add generalizations to the end of the article, if you believe they are notable (and have appropriate sources). The two books (Pedersen, and Reed-Simon) do consider more general spaces (see chapter 2 "Banach spaces" in Pedersen, and chapter 5 "Locally convex spaces" in Reed-Simon), but nevertheless, both books treat unbounded operators on Hilbert spaces only. I conclude that they find such generalizations non-notable. Boris Tsirelson (talk) 15:55, 18 April 2009 (UTC)
As far as I know, closed densely defined operators are typically studied in the context of Banach spaces (hence, my above comment). Maybe they restricted the cases to Hilbert spaces for pedagogical reasons. We don't have to do that here in Wikipedia. Also, I'm not sure if you are opposing the change or not. As I said below, I think we should merge densely-defined operator and closed operator to this article and then have a section on a closed densely-defined operator and symmetric and self-adjoint operators. -- Taku (talk) 23:39, 18 April 2009 (UTC)
I am not opposing the change provided (once again) that they are notable, and you have appropriate sources. Boris Tsirelson (talk) 05:21, 19 April 2009 (UTC)

One more thing. Does anyone think merging densely-defined operator and closed operator to here makes sense? The two articles seem to duplicate materials (for obvious reasons). -- Taku (talk) 11:18, 18 April 2009 (UTC)

About the merge, I am neutral. In fact, I see, you are already doing so, gradually. Boris Tsirelson (talk) 05:21, 19 April 2009 (UTC)
"<!-- Actually the kernel of any closed operator is closed? -- Taku -->" – Yes, you are right; the proof is immediate. Boris Tsirelson (talk) 05:28, 19 April 2009 (UTC)
Several comments: I think it is far too arbitrary to merge the two concepts of densely-defined and closed. Also, this article is very detailed, and trying to pack more information into this article would begin to make it too difficult to absorb.
I think a better idea would be to start a new article titled something like "Types of operators" (if there is not already such an article). In that article there could be *brief* definitions of the major types of operators like closed, densely-defined, symmetric, adjoint, continuous, bounded, and unbounded, etc., plus maybe one example of each kind and the most basic properties of each one.
For more detailed information such as is found in this article, there can be entire articles on the most important kinds of operators -- if anyone feels like writing such an article.
One last thing: I have the impression that people think "densely-defined" means "defined on a dense subspace that is not the entire space. I don't know whether that is the custom in operator theory, but if it is, then "densely-defined" has a different meaning from its most obvious meaning (defined on a dense subspace), since that "obvious" meaning would include the entire space, since that is dense in itself.

IF this is the custom, contrary to the intuitive meaning, then this should be explicitly stated wherever "densely-defined" is "defined".Daqu (talk) 06:29, 5 September 2010 (UTC)

"contrary to the intuitive meaning"? For a non-mathematician, maybe. A remark is added. Boris Tsirelson (talk) 08:22, 5 September 2010 (UTC)

What is the purpose of the following statement: If T : B1B2 is closed, densely defined and continuous on its domain, then it is defined on B1. Surely if an operator is continuous then it is always bounded, and this article is devoted to unbounded operators only… ?  // stpasha »  09:48, 6 September 2010 (UTC)

Also since the article is on unbounded operators, the “Defintion” section should include the definition of the unbounded operator.  // stpasha »  09:57, 6 September 2010 (UTC)

Did you read the lead? Yes, "the article is on unbounded operators", but still, '"unbounded" should be understood as "not necessarily bounded"'. The definition is given. When needed, you may say "... and not bounded". Maybe you do not like it, but this is the usual terminology (here as well as in other parts of math; for example, the theory of nonlinear operators is not a theory that assumes nonlinearity; rather it is a theory that does not assume linearity). About your former question: yes, it is bounded on its domain; but the statement is that its domain is the whole space. It is here because (1) it is about operators defined not just everywhere, (2) it is about closed and desely defined, and (3) it can be reformulated: if the domain of a closed densely defined operator is not the whole space then the operator is not bounded (equivalently, not continuous). Boris Tsirelson (talk) 11:02, 6 September 2010 (UTC)

I disagree[edit]

"Since the domain of a bounded operator is closed, the densely-defined operator (but not necessarily defined everywhere) is necessarily unbounded." Really? I fail to understand it. First, why "the domain of a bounded operator is closed"? If we accept the convention that "in the special case of a bounded operator, still, the domain is usually assumed to be the whole space" then it is not just closed. If we do not then it need not be closed (just restrict a bounded operator to a non-closed subspace). A true statement would be "A closed densely-defined operator whose domain is not the whole space cannot be bounded". However, is it a notable statement? Boris Tsirelson (talk) 16:08, 18 April 2009 (UTC)

Ok, you're right. We first have to clarify the convention. It is not clear (I mean, the article doesn't make it clear) what a bounded densely defined operator is. Is a bounded operator defined everywhere by definition or by a proof? (I have to look at books.) -- Taku (talk) 01:47, 19 April 2009 (UTC)

"or equivalently, the domain of the operator is a Banach space in the graph norm" – if you want to say so, probably you have to say also, what is the graph norm. And by the way, what is denoted by \operatorname{gra}T  ? (I guess what, but it is not explained.) Boris Tsirelson (talk) 16:19, 18 April 2009 (UTC)

Yes, we probably should clarify them. -- Taku (talk) 23:10, 18 April 2009 (UTC)

"Consequently, the most important class of closed operators are densely defined ones" - what does it mean? Maybe you mean "the domain of a closed operator is not closed unless the operator is bounded"? Do not forget that the whole space is also a dense set. Boris Tsirelson (talk) 16:23, 18 April 2009 (UTC)

I meant that closed operators with domain that isn't dense are uninteresting. I thought it was clear. -- Taku (talk) 23:10, 18 April 2009 (UTC)
Or maybe you really mean that "closed operators with domain that is close are uninteresting"? The closed graph theorem is irrelevant to denseness of the domain, it is relevant to its closedness. Boris Tsirelson (talk) 05:36, 19 April 2009 (UTC)
But if the operator is only densely defined (i.e., not necessarily everywhere defined), the closed graph theorem doesn't apply because the theorem relies on the completeness. -- Taku (talk) 11:15, 19 April 2009 (UTC)
The closed graph theorem applies to a closed operator if and only if its domain is closed. It does not matter, is the domain dense or not. That is, it does not matter, is the domain the whole space or another closed subspace. Boris Tsirelson (talk) 13:10, 19 April 2009 (UTC)
You're correct. I guess what I wanted to mean is: "closed operators defined on a Banach space are uninteresting (from the point of unbounded operators) because they are bounded by the closed graph theorem". This sounds more accurate :) -- Taku (talk) 22:44, 19 April 2009 (UTC)

"A densely-defined operator T admits the adjoint operator" – do you assume that the reader knows what is called "the adjoint operator"? Boris Tsirelson (talk) 16:27, 18 April 2009 (UTC)

Why don't we just clarify this too then? -- Taku (talk) 23:10, 18 April 2009 (UTC)

"By the closed graph theorem, every closed operator that is defined everywhere on its domain is bounded" – what does it mean? Every operator is defined everywhere on its domain, just by the definition of the domain. Boris Tsirelson (talk) 14:31, 19 April 2009 (UTC)

"defined everywhere on its domain" -> "defined everywhere". -- Taku (talk) 22:30, 19 April 2009 (UTC)

"One consequence of this definition is that a sum and product of two unbounded operators may not have the same domain. In particular, the commutator of two unbounded operators (and so the commutation relation) is not well-defined in general." – Why such a difference between sum and product on one side, and commutator on the other side? Each one (sum, product, commutator) can be defined, and its domain consists of all vectors where the formal expression make sense. Each one can appear to be trivial (the domain containing only zero) or nontrivial. And what does it mean, "may not have the same domain"? D(S)≠D(T)? D(S+T)≠D(S)? D(S+T)≠D(T)? D(S+T)≠D(ST)?? Or what? Boris Tsirelson (talk) 04:38, 20 April 2009 (UTC)

Also I doubt that this phrase should be in the "Definition" section. Boris Tsirelson (talk) 04:40, 20 April 2009 (UTC)

Yes, the part is unclear. But I know that the sum (or product) of unbounded operators is very problematic. Since how to define sum or commutator isn't straightforward, it makes sense to discuss it in the "definition" section. (I have to go now, I should be able to edit the article later.) -- Taku (talk) 11:56, 20 April 2009 (UTC)

I also disagree with an edit by User:Algebraist. Before, when clicking on some item in the "Notes" section I got the corresponding item in the "References" section. Now I do not! For most cases, instead I see "References" disappearing from the screen! Quite frightening. Well, I see already that the section still exists, it only becomes temporarily invisible. Thus, it seems to me, this edit was destructive. Boris Tsirelson (talk) 18:39, 13 June 2009 (UTC)

Oops, I am sorry. It is a problem of Firefox2 and two-column "Notes" section. Not related to the edit of Algebraist. Boris Tsirelson (talk) 19:32, 13 June 2009 (UTC)

Symmetric operator[edit]

A fair amount of materials on symmetric operators seem to duplicate Extensions of symmetric operators. Is it time for us to start symmetric operator? (Since I'm not an expert on the extension issues related to symmetric operators (there are tons, according to Rudin, for example), I'm not uncomfortable doing that myself. But I do believe, rather strongly, that this article should focus on the general properties of a unbounded operator (whence my extensive revision.) -- Taku (talk) 01:07, 29 June 2009 (UTC)

This article is *seriously* deficient[edit]

Any article on unbounded operators that (as far as I can tell) fails to give even one concrete example of such an operator is seriously deficient. Sorry, just referring casually to the foundations of quantum mechanics doesn't even begin to remedy this glaring omission. (I am not the person to fix this, because I am no expert on the subject of this article by any means.)

Also: If the underlying linear space is assumed to be Hilbert space (which one, by the way, or do you mean any Hilbert space?), then a much better title for this article would be Unbounded operators on Hilbert space.

I am constantly amazed by some contributors to Wikipedia who may be quite expert in the subject of an article they edit, but haven't the vaguest idea of what constitutes a good encyclopedia article.Daqu (talk) 20:15, 29 August 2010 (UTC)

Would you please explain more, what constitutes a good encyclopedia article? I can add an example, but I'd prefer to see the whole picture; maybe there are a lot of even more serious deficiencies? Maybe you have an example of a good article written by you.
About examples: do you mean, an example as simple as possible? Or rather, a really useful (but not simple, regretfully) example?
About Hilbert space: see the discussion on top of this page.
Boris Tsirelson (talk) 20:42, 29 August 2010 (UTC)
I will mention three of the qualities that a good encyclopedia article should have: 1) It should be clear. 2) It should address itself to readers with various levels of knowledge, including those who may know very little or nothing about the specific subject. 3) It should be accurate.
In summary, a good article will have the function of teaching the reader about its subject, just as a classroom teacher might do.) Of course, technical articles like this one need to assume a certain level of familiarity with mathematics, but there is no need to assume more than is required to understand a brief definition of the article's subject.
Both of the principles 1) and 2) dictate that an example be given in any article about a specific kind of mathematical object. And that either such an example be self-evident as to why it is an example of the object, or that a clear explanation be provided.
I think in another article that links to this one, on the space of square-integrable functions on the reals, one example of an unbounded operator is given as d/dx, and another as multiplication by x. I am not knowledgeable in the area of operators, so I'm not the person to add the example(s) here.Daqu (talk) 19:32, 1 September 2010 (UTC)
OK, an example is inserted. Is it still "*seriously* deficient"? Boris Tsirelson (talk) 07:13, 2 September 2010 (UTC)
Thank you! The article is now many times better than it was before.Daqu (talk) 06:09, 5 September 2010 (UTC)

Really confusing[edit]

I don't understand this. What's more, I'm not the only one. Various articles on operators claim Quantum Mechanical "observables" are bounded operators while others claim they are unbounded. It is obvious that either one claim is wrong, or from what I can gather from this almost impenetrable article, both are under-specified. The lede allows that an unbounded operator may be bounded ("not necessarily bounded" means "may be bounded"), and has all sorts of rubbish about what it "usually" refers to or doesn't refer to, except when it doesn't or when it does. I conclude that the author of this isn't skilled enough in the subject to clearly articulate what the definition actually is. Look, F:X→Y MEANS the function F maps (all) elements of the set X to the set Y. This is the default meaning, and should not be obliterated by silliness such as d/dx:C[0,1]→ C[0,1] because the domain is under-specified; it is NOT C[0,1], it IS, by definition, specifically, C¹[0,1]. The entire article seems to be based on forcing ambiguity into the mathematical terminology. I understand that if B ⊂ A and C ⊂ B then C ⊂ A so if F:C→X is true, then F: A → X is true iff a ∈ C for each element in the domain of F. Which is NOT A, it is C. (assuming C < A ). My point is that the treatment of the sets (spaces, if you want) is virtually ignored when they are actually key in explaining what an unbounded observable is. Its claimed that von Neuman used them to rigorously define the foundations of Q.M. while here they seem to be more "I know it when I see it, but I can't explain it". If the terms "bounded operator" and "unbounded operator" depend on the specific space (subspace) being referred to, then clearly that needs to be mentioned in the lede. Vascillating between speaking about them in context of some larger space and simultaneously in the context of their domain is just sloppy (and confusing). Why is it the case that (d/dx):C[0,1]→C[0,1] is "well-defined" ? G = |x| has no "well-defined" value for dG/dx at x=0. And it is, as far as I can tell, meaningless to claim that there exists "some subspace" (possibly a single point) where if it is true for that, then it is true for the space. Also, I don't understand the use of fn in fn(x) = sin(2πnx) why is the "n" present? and assuming it is a parameter, why bother including it? If it means something different, then make that explicit. (The example given claims that if T=2 then dfn/dx → ∞ but that has nothing to do with D(x) and everything to do with D(n).) Very confusing. (talk) 16:01, 21 June 2014 (UTC)

Well, the article was started by me. But it does not mean that I am the author of that terminology. The terminology existed long before. We may hate that terminology; but we cannot annul it. If in doubt, look at the books cited (and many other books).
The above is about mathematical literature. Now about physics literature. You see yourself that "Various articles on operators claim Quantum Mechanical "observables" are bounded operators while others claim they are unbounded." Let us not blame each other for all that; we are not guilty. Such is the life, and all the WP article can do is, to honestly inform the reader.
If you have a question about the existing terminology, I'll try to help. But if you just protest, I am of no help.
Now about f_n(x) = \sin (2\pi n x). This is an absolutely clear, and very usual, definition of a sequence of functions. I wonder, what is your problem with it? Did you ever deal with a sequence of functions before?
Boris Tsirelson (talk) 16:55, 21 June 2014 (UTC)