Talk:Operator

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[edit] WHAT IS GOING ON AT THIS ARTICLE???

This page is really confusing. It never says what an "operator" is, and hence it wanders all over the place, on the one hand it appears to treat "operator" as synonymous with "linear transformation" on vector spaces, with the "special" case of operators from function spaces to function spaces as some special case. (I find this distinction unwarranted...linear operators from R^n to R^m are really the same thing, since vectors in R^n, e.g., are really just "functions" from the set {1, 2, 3,..., n} to R, they are the discrete analogue of continuous function spaces.) But later on in the article, there is mention of some combinatorics functions, and the factorial function...I don't see how this fits in with the previous stuff...I don't see how factorial is "linear" in any way, so now the article seems to be embracing a much wider conception of "operator" than just "linear transformation", indeed, at certain points in the article, the impression is given that "operator" is virtually synonymous with "function" or "mapping" of sets, regardless of any linear structure (or any other structure) present. And then the presence of some computer science usage of the term just muddles things up completely with respect to any mathematical meaning intended. (I won't mention minor stuff like the "a" in the definition of linear belongs outside the "f", not inside "f".) It seems that much of this is already done much better at linear operator, and moreover, there needs to be some kind of discussion just what precisely is to be meant by "operator" at this article. It's kind of a generic term that's used in a lot of areas of math, so it may not have some fixed meaning depending upon what you happen to be reading or working on. Revolver 10:25, 5 Apr 2004 (UTC)

I agree. I vote this page is split up and turned into a disambiguation page. Looks like an icky job, though. Lupin 10:36, 5 Apr 2004 (UTC)
I'll go over this again. If I can't tighten this up, then maybe it needs to be taken out and shot in the head ;) Dysprosia 10:55, 5 Apr 2004 (UTC)
The primary problem is that the subject of the article ("operator") is never clearly defined, so the article ends up treating a variety of (IMO) unrelated topics, each of which could be moved and expanded upon on other pages making it more clear what the specific usage is. Then maybe this could become a disambig page, although mostly for mathematical, also others...one link to linear transformation, others to unary/binary operations, function, integral transform, physics, etc., etc. Not everything in this article is badly written, but it doesn't all belong in the same place, is all I was saying. Revolver 11:09, 5 Apr 2004 (UTC)

Some suggestions. Firstly, there is some need for a discussion of what one means by an operator in terms of type: namely that operator often means 'accept a function and return a function'. I know it doesn't always mean precisely that; but operator the concept is up there with function space, functional and higher-order function as big signposts to the use of abstraction in mathematics (and not only there ...).

Second, it would help to add 'main article' links, so that for example the Fourier transform section is rather obviously about 'what style of operator is the FT?', rather than 'all you ever needed to know about FT'.

Charles Matthews 11:11, 5 Apr 2004 (UTC)

I did make the main article links rather implicitly the first time round. Thanks for pointing this out, I'll make this clearer. Dysprosia 11:21, 5 Apr 2004 (UTC)

Ok, I think I removed most of the crud and silliness in this article. I hope it's better now :) Dysprosia 11:42, 5 Apr 2004 (UTC)

I think this is now going in the right direction. More from me, but later.

Charles Matthews 11:54, 5 Apr 2004 (UTC)

Yes, I agree it's getting better. One comment, just for the moment -- there is a section on linear operators, and then other sections on differential operators, integral operators, transforms, etc. But many of these are themselves linear operators on appropriate vector spaces, i.e. they are special cases of the "linear operator" example. Placing them outside the "linear operator" section may give the impression they aren't linear, or at the least, it doesn't draw attention to this fact. Maybe they could be incorporated into the linear operator section, or given as subsections? Revolver 22:53, 5 Apr 2004 (UTC)
You're quite right! I'll try and fix this. Dysprosia 22:57, 5 Apr 2004 (UTC)
I see there's a small mention of this in the section on linear operators, but it could easily be overlooked by someone. Revolver 22:59, 5 Apr 2004 (UTC)
The problem is that we may start running in to too deeply nested subheadings. Subheadings are nice, but it could get a little complicated. (Re the linear operator and the "a" thing, didn't I have it right the first time? :) Dysprosia 23:06, 5 Apr 2004 (UTC)
Yeah, I was thinking about that (getting too nested). I'm sure there's a way to make it more clear without nesting too much. As for the "a", actually, no it wasn't right either the original way or after your change. The linear operator is Q, not f, i.e. it is not f itself that is linear, but Q (f might be the sine function, for instance). It may be more clear if you see it without the x arguments:
  • Q(f + g) = Qf + Qg
  • Q(af) = aQ(f)
This is purely subjective nit-picking, but the Q notation seems weird -- the usual symbol for a generic linear transformation is usually something like T or L. Revolver 23:53, 5 Apr 2004 (UTC)
I know - I had that originally, and in my cleaning up I must have mistakenly changed it to the other way round :) Dysprosia 03:04, 6 Apr 2004 (UTC)

I've made some further additions and changes round. There is some more to do, but I think the balance of the article has been improved, in favour of telling the reader what an operator is (or might be). One thing more would be to decide about div, grad and curl. The high-brow mathematical view is that they manifest two operators (exterior derivative and Hodge star operator). This could really do with being on its own page, though.

Charles Matthews 15:41, 6 Apr 2004 (UTC)

More moves, some minor cuts - looks more like a definitive job now, IMO. Sorry about the clash of heads, Dysp - don't think anything of yours got lost, when I repasted my work. The intro is much more upfront now, I think; at a cost of being more abstract, though. Charles Matthews 11:38, 13 Apr 2004 (UTC)

I made a number of changes to the formatting here. Originally I changed the symbols from blue because I (perversely?) expected the operators to be links. I suppose I've been properly propagandized in re: websurfing: blue is for links, purple for visited links.... So then I tried bolding them, with no success, then switched them to red instead, but now they're distracting from the text itself. So I'm restoring them to blue and washing my hands of it.  :-)

This is definitely a stub. It hardly hints at most of the things mathematicians think of upon hearing the word "operator". Michael Hardy 19:30 Jan 22, 2003 (UTC)

Agreed. Operation is pretty bad too -- Tarquin 19:32 Jan 22, 2003 (UTC)

I am rewriting this article greatly, if anyone is interested. Could this be checked for accuracy once I'm done? Dysprosia 05:39, 12 Aug 2003 (UTC)

Will do, just let me know when you've finished as it seems to be better to have just one person working on a thing at one time.  :-) BTW, I thought additive operators had to be comuntative, though I'm not absolutely certain. -- Daran 06:21, 14 Sep 2003 (UTC)
It's kind of in a beta state right now. Most of the stuff should be right, but it's not complete.
However if you see anything blatantly wrong, change it! Be bold! :) Dysprosia 06:22, 14 Sep 2003 (UTC)
To be quite honest, I don't even want to look at it right now, for fear of being distracted from the other things I'm working on.  :-) Thanks for the welcome, BTW. -- Daran 07:09, 14 Sep 2003 (UTC)

Am I the only one who thinks it is an abomination to say that an operator is a symbol? Usually an operator is thought of as a mapping from a space to itself or to another space. Michael Hardy 01:28, 4 Jan 2004 (UTC)

Second! wshun 01:35, 4 Jan 2004 (UTC)
Second second! Herbee 16:06, 2004 Mar 1 (UTC)
Define what you mean by a "symbol"? Dysprosia 02:33, 2 Mar 2004 (UTC)

I feel like it's probably awkward to be discussing computer science operators and math operators in the same article. though they're related, the CS bit really isn't at all relevant to someone reading about operators in mathematics. the operator (disambiguation) page is fairly substantial. perhaps the CS bit should be split out and operator should become the disambiguation page. I'd be willing to do this work if it should be done. SeanProctor 03:07, 22 May 2004 (UTC)

I don't think things have to be segregated that way. It's a bit like saying we can make one article per POV, rather than a NPOV article. Charles Matthews 08:35, 22 May 2004 (UTC)

if you were replying to my comment, I don't understand your point. where did the "have to be" come from? I never said that. I was saying basically what you said in the second half, except I think the CS/math bit is just a mildly related topic, not a different POV on the same topic. see function SeanProctor 21:07, 31 May 2004 (UTC)
No one explicitly opposed this, so it's done. SeanProctor 02:09, 27 Jul 2004 (UTC)

[edit] When to use calligraphy notation

When to use F\, and \mathcal{F} should be covered in this article. One is a function and one is an operator, but this article says that an operator is just a special type of function... - Omegatron 02:35, Sep 19, 2004 (UTC)

I'm sorry I was confused by the Fourier transform article. (Talk:Fourier_transform#F_notation)I will add a bit. - Omegatron 02:43, Sep 19, 2004 (UTC)

[edit] Properties?

This article seems pretty extensive and covers many interpretations of an operator but I was curious if the addition of a Basic properties section might be helpful. For example:

The following are properties of operators, for operators \hat{S} and \hat{T} and a function f:
(\hat{S} + \hat{T}) f = \hat{S} f + \hat{T} f
(\hat{S} - \hat{T}) f = \hat{S} f - \hat{T} f
\exp(\hat{S}) \equiv \sum_{n=0}^\infty \frac{\hat{S}^n}{n!} = \hat{1} + \hat{S} + \frac{\hat{S}^2}{2!} + \frac{\hat{S}^3}{3!} + \cdots
where \hat{1} is the identity operator.
  • ...

I know such properties are true for the definition of operator that I worked with in my physics/math coursework which applies to more than just linear operators but I'm not sure how universal these properties are. Maybe someone with more expertise knows? In any case, this seems like worthwhile addition to this article. What do you think? -Dan Granahan 17:46, 21 July 2005 (UTC)

These are properties of some special types of operators but not of operators in general. Zaslav 08:08, 25 June 2006 (UTC)

[edit] confusing passage

The following passage needs to be more pedagogical before it is placed into the article. It especially does not belong at the beginning in its current state - Gauge 00:36, 8 January 2006 (UTC)

The arguments to operators are sometimes called operands. The most frequently met usage of operators is a mapping between vector spaces; this kind of operator is distinguished by taking one vector and returning another. For example, consider an enlargement, say by a factor of √2; such as is required to take one size of DIN paper to another. It can also be applied geometrically to vectors as operands.

[edit] A Question of Linear Operators

I have a question of Physics about Math that is on [1]. Why does

\mathcal <P^2>={\Delta}P^2+<P>^2 ?

--HydrogenSu 09:48, 26 January 2006 (UTC)

It's solved somewhere. Thanks for care-me boys.--HydrogenSu 11:41, 27 January 2006 (UTC)

[edit] Function = Operation = Operator ?

It is a nice distinction to keep the concept of function, which typically works with numbers, separate from the concept of operator, which typically works to turn one operation into another.

In mathematics, differentiation and indefinite integration are operators. They operate on functions, turning them into other functions, analogous to functions on numbers turning them into other numbers.

This distinction makes sense as long as infix functions such + are considered functions and not operators. This is reasonable, because what the function + does is add its two arguments, one of which appears on the left of the function symbol and the other on the right. I apologize here for conflating the notation with what it represents.

To people who know and love the mathematical notation and computer language APL, a function is something that takes one or two, or occasionally zero arguments, and usually (but not always) returns a result. In contrast, operators turn functions into other functions. For example, the reduction operator, written with the slash, requires a function on the left and an array on the right. In the simplest case, the array is a vector, and the function is applied as if inserted between every adjacent pair in the vector. Thus, +/ 3 5 2 6 is evaluated as 3+5+2+6. In other words, +/ is APL notation for array summation, usually represented in mathematics with Σ (Sigma, for sum) and ×/ is APL notation for array product, typically represented in mathematics with Π (Pi, for product). But APL generalizes this in at least two important ways: First, any primitive scalar dyadic function can be reduced. For example the "and" and "or" functions can be reduced. Second, mathematics defines Σ and Π only on vectors, not on matrices or higher arrays, but APL defines reduction on arrays of arbitrary rank. Anomalocaris 09:23, 12 December 2005 (UTC)

While the distinction may be nice ("nice adj. nic·er, nic·est: [...] 5. Overdelicate or fastidious; fussy.") :) the fact is that many mathematical texts do not follow this distinction; and I think Wikipedia should follow actual use, not prescribe it. Lambiam 14:55, 6 March 2006 (UTC)
Wikipedia's co-founder, Jimmy Wales, has called Wikipedia "an effort to create and distribute a multilingual free encyclopedia of the highest possible quality to every single person on the planet in their own language.
I think this article doesn't show the highest possible quality. It mixes different concepts, it confuses a thing and its symbol. This practice has been rejected by the greatest mathematicians like Frege (who spent half of his life combatting against this practice) or Bertrand Russel. It is not "prescription" when xou don't follow an obviously false custom.-
This custom can cause problems not only in mathematics (mainly, in the foundations of mathematics), but in information science, too. For example, + is one, concrete operator. But it is not one, concrete function for example in a strongly-typed programming language, like Pascal. See, it signs addition of real-type data and addition of set-type objects. ♥♥♥: Gubb  ✍    2006. May 20 15:28 (CEST) 15:28, 20 May 2006 (UTC)
"In mathematics, differentiation and indefinite integration are operators. They operate on functions, turning them into other functions, analogous to functions on numbers turning them into other numbers."
Yes. But we use this term only in mathematical analysis, to be exact, in operator calculus (or operator analysis). But in this meaning "operator" is an exact term for some functions, what are mapping from a function space to the same function space. But it is false to say "In mathematics, an operator is a function that performs some sort of operation on a number." (This sentence is ungainly for other reasons, too. For example, says that "operator" is a function that performs operations ... but what does it mean "operation"? You can see, it is also a function. So operator is a "function that performs a function"). In general, it is not true in mathematics operator is a function. This meaning can be used only in information science (computer programming etc., where, unfortunately, this meaning is really in use sometimes). So in analysis "operator" is an exact term (like, for example, "functional", what is a functon mapping from a function space to the set of real numbers). This is a special meaning of "operator" in mathematics, so should be detailed in the article "operator (analysis)"". The word "operator" has a different meaning in mathematics, too, used in linear algebra and abstract algebra; when given two disjunct sets, the O "operator domain" and a set A; and given an "external operation", a function O×A->A (producting vectors with scalars, polinomials above a field with the elements of the field etc.), then we call elements of the set O as "operators" (see operator groups, or see modules). This two meanings are not independent, sith when given the function space A, an ω operator mapping from A to A, is an element of the set AA:=O, and we write ωf=g instead of ω(f)=g; so we can define an external operation F:O×A->A; F(ω, f)=ωf=g iff ω(f)=g. But I would risk that statement, except these two partial and hiperprofessional meaning of the word, no serious mathematician would say anything else about "operator", unless that "it is a symbol for operations". So, what you mean, that is operation and not operator. ♥♥♥: Gubb  ✍    2006. May 21 10:31 (CEST) 10:31, 21 May 2006 (UTC)

Here is a supposed reason for using the term "operator" instead of "function":

  • To draw attention to the fact that the domain consists of pairs or tuples of some sort, in which case operator is synonymous with the usual mathematical sense of operation.

I removed this from the article because, in my experience in mathematics, this is never the reason for calling a function an operator. If anything, it's somewhat the opposite, since many if not most kinds of operators are unary (all differential, integral, and linear operators, for example). Furthermore, the terms "operator" and "operation" are not synonymous, as they are used in different ways. Zaslav 08:49, 25 June 2006 (UTC)

I agree (have not heard anything before like that). ♥♥♥: Gubb  ✍    2006. July 6 12:26 (CEST)

[edit] Operators in calculus

The article refers to the indefinite integral operator and then shows the integral sign with 0 as the lower bound and and t as the upper bound. Isn't an indefinite integral represented by an integral sign without lower and upper bounds? —The preceding unsigned comment was added by 67.101.96.81 (talk) 20:20, 9 April 2007 (UTC).