Talk:Operator (mathematics)

WHY NO MENTION OF ELEMENTARY OPERATORS?

Most of wiki users are not specialized. Why not start with a simple discussion of familiar arithmetical operators and some algebra? Mrdthree (talk)

Because arithmetic operations are not operators. Kallikanzarid (talk) 06:17, 5 April 2010 (UTC)

Operation and operator are different concepts but you seem to be using them in the same way. I would say that arithmetic operators are most definitely operators. My understanding of an operator is that it is a symbol with an associated syntax that represents a function. For example, addition of two elements (over the reals e.g.) would constitute a function but + is an operator. Operators exist for the purpose of making it easier to conceptualize and perform algebraic manipulations. For example, in operators you can express associativity as (a+b)+c=a+(b+c), where as in functional notation the expression add(add(a,b),c) = add(a,add(b,c)) makes the rule less clear. 174.91.192.80 (talk) 12:05, 12 October 2013 (UTC)

Shouldn't there be more elucidation of the commonly encountered cases where the vector space in question is always some broad space of functions, such as for the derivative operator? Moreover, the familiar arithmetic operators are operators in the sense of this article, just for trivial choice of vector space. But saying so seems tautological, since familiar arithmetic operations are already used in the definition of vector space.. alternatively, if "arithmetic operations are not operators" (or at least are defined in terms of some more primitive concept) then still shouldn't the lead be the place to mention this and to link to whatever it actually is then that arithmetic operators are defined as? Cesiumfrog (talk) 23:17, 19 December 2010 (UTC)

"Shouldn't there be..." Isn't it there in the examples section?

"Moreover, the familiar arithmetic operators are operators in the sense of this article..." You could say that $+:\mathbb {R} \times \mathbb {R} \to \mathbb {R}$ is an operator from $\mathbb {R} ^{2}$ to $\mathbb {R}$, but this isn't a very smart decision, because

1. As you pointed out, "+" is already in the definition of the vector space as an operation. Moreover, most questions of interest about such operations and their generalization are studied outside the setting of vector spaces and their operators (see group (mathematics), ring (mathematics), field (mathematics)).
2. The study of the properties of the operators are driven by completely different questions, ideas and problems than the study of arithmetic operations and their generalizations. The most practically useful operators are actually linear: they preserve vector space operations; trying to study "+" as a linear operator wouldn't be very useful. Actually, $x\mapsto \alpha x$ is a little bit interesting example, but it is still very trivial; and when you're trying to multiply two vectors, you're most likely already dealing with algebra, which is usually studied on its own.

In short, while technically you can call arithmetic operations (but not their more interesting generalizations) operators, it is not very meaningful to do so. This is why it is benefetical to separate the notions of operation and operator and make sure that links in Wikipedia lead people to the setting that is most appropriate for studying their problems. Last but not least, studying binary arithmetic operations in the setting of vector spaces and their operators requires one to know what the direct product of vector spaces is, and it is way above what you can expect from someone with only school level of exposure to math.

"...then still shouldn't the lead be the place to mention this..." It is, there is a link to operation (mathematics). — Kallikanzarid^talk 05:34, 20 December 2010 (UTC)'

Case in point: The page Operand says: "The operand '3' is one of the inputs (quantities) followed by the addition operator", and the word "operator" is a link to this page, which isn't any help at all if you want to read about the likes of +, -, etc. 147.1.11.12 (talk) 23:35, 19 August 2014 (UTC)

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)

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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.  :-)

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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)

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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)

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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)

Hi!

“In basic mathematics, an operator is a symbol or function representing a mathematical operation.” An extremely wide 'definition' I'd say (being a linguist)... "Symbol or function"?? Oooo... JLincoln (talk) 15:54, 1 July 2012 (UTC)

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)

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:

• The sum and difference are given by

$({\hat {S}}+{\hat {T}})f={\hat {S}}f+{\hat {T}}f$

$({\hat {S}}-{\hat {T}})f={\hat {S}}f-{\hat {T}}f$

• ...
• The exponential of an operator is defined by its power series

$\exp({\hat {S}})\equiv \sum _{n=0}^{\infty }{\frac {{\hat {S}}^{n}}{n!}}={\hat {1}}+{\hat {S}}+{\frac {{\hat {S}}^{2}}{2!}}+{\frac {{\hat {S}}^{3}}{3!}}+\cdots$

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? -Vanished user 1029384756 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)

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.

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)

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 A^A:=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)

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).

Definition is too narrow

I must take issue with the introduction to this page. When I read it, I was shocked. I have spent decades reading math books of all sorts, and the definition went all that my experience tells me.

An operator is a function, of one or several variables. The inputs do not have to be functions, nor do the outputs. I don't even think it is fair to say that "usually", in mathematics, the inputs and outputs are functions. In my experience, operator has always been almost synonymous with function. The only distinction I think you can make is a tendency for operators to refer to functions whose inputs and outputs are drawn from the same set, but again, not necessarily a set of functions. To back me up, here is a link to the definition at the American Heritage Dictionary:

[2]

I thought I was going crazy, seeing the definition currently on the page. Will anyone else back me up here?

It sounds to me like the current definition might come from a field other than mathematics. If so, then there needs to be some sort of disambiguation here. —Preceding unsigned comment added by B2smith (talk • contribs) 23:07, 9 December 2008 (UTC)

Notation of a symbolic operator

The article needs to say how an operator without a unique symbol is commonly denoted, as with x for a variable and c for a constant. LokiClock (talk) 13:20, 17 February 2010 (UTC)

It depends on the context. Linear operators are often T, binary operators can be a lot of things depending on the context (group? abelian or not? ring addition/multiplication? something else?). The idea of an operator is very generic; what you're asking is like asking what symbol is typically used to denote an algebraic structure. Usually G or H or K for groups, R for rings, F or K or k for fields, N for normal subgroups, I for ideals, etc., etc. —Simetrical (talk • contribs) 19:46, 17 February 2010 (UTC)

Yes, I know it's a general thing, but that doesn't mean people don't discuss things that general. I've seen • used before in this context. LokiClock (talk) 20:21, 17 February 2010 (UTC)

· is used for binary operators in some cases, but not for unary operators – I've never seen "·x" or the like. I don't think the article needs to specifically mention this, anyway. —Simetrical (talk • contribs) 19:51, 18 February 2010 (UTC)

It would be useful to have the information as reference when someone sees the use of notation, or when someone wants to use a typical or de facto standard symbol for discussing, e.g., the properties of arbitrary operators. So pretty much the same reasons you'd include such information on any frequent notation. LokiClock (talk) 14:40, 19 February 2010 (UTC)

Finitary relation uses Lu, Luv, Luvx, etc. for an n-ary operator L. Is this or a similar notation common? ᛭ LokiClock (talk) 15:11, 25 June 2010 (UTC)

I reworked the introduction

Please check the punctuation (I'm not native English speaker), and fix Continuous operator redirecting to Bounded operator---these terms are distinct!

Also, PLEASE refrain from editing this page if you confuse an operator with an operation, or if you don't know the difference between a function and a mapping. Kallikanzarid (talk) 06:16, 5 April 2010 (UTC)

About function and mapping: my bad :) —Preceding unsigned comment added by Kallikanzarid (talk • contribs) 13:15, 22 November 2010 (UTC)

Split the article?

IMO this article needs to be split, mainly to protect Operator (mathematics) from well-intentioned vandalization by overconfident programmers <_< Kallikanzarid (talk) 06:28, 5 April 2010 (UTC)

Ok, since there is a link to a disambiguation page, to hell with gentle manners! >_< I'm throwing all unrelated stuff out. Kallikanzarid (talk) 12:07, 22 November 2010 (UTC)

I'm starting a revamp of the article

This article is a mess, and the only way we can EVER hope to fix it is to remove all stuff relating to Operator (programming) from this article. It is justified, because the distinction is made right under the header and the link to the disambiguation is given. Kallikanzarid (talk) 10:46, 22 November 2010 (UTC)

I removed all the confusing stuff and layed out some resemblance of good article structure (that will probably not be able to stand open scrutiny :)). I left a lot of comments that, should you be fine with such a structure for a while, will serve as a guide towards extending the article. Let's put some effort into this, people — this is an article on operators, christsakes! Kallikanzarid (talk) 12:06, 22 November 2010 (UTC)

Rename the article?

It would be more logical for this page to be called Operator (mathematics). There are already Operator (programming) and Operator (disambiguation). —Preceding unsigned comment added by Kallikanzarid (talk • contribs) 12:51, 22 November 2010 (UTC)

Requested move

The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the move request was: page moved. Vegaswikian (talk) 21:46, 16 January 2011 (UTC)

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Operator → Operator (mathematics) — This article is about operators in mathematics. There are good reasons why it should be moved:

• the word 'operator' has many meanings, and the meaning used on this page is technical and is unlikely to be sought for by anyone searching for 'operator'.
• The ambiguous name leads to many incorrect links (I have resolved some of these, but not all yet). The worst problem is confusion with operator (programming) which lead to misguided edits by programmers.
• There is a confusion of terms 'operator' and 'operation' in mathematical context. I'm afraid there is no single best way to address it, but redirecting operator to a dab page would help at least to some degree. It is also possible to move this page to operator on a vector space or the like and make operator (mathematics) a page that deals with this confusion and possibly even provides a historical perspective. — Kallikanzarid^talk 15:56, 9 January 2011 (UTC)

• Support. Per Kallikanzarid. Paul August ☎ 21:10, 9 January 2011 (UTC)
• Support. Per nom. The mathematical meaning is just one of several, and it is not clear that a vast majority of people would be searching for the mathematical meaning. (In fact, this seems unlikely.)TimothyRias (talk) 22:35, 9 January 2011 (UTC)

  Of the 19K who hit the page in November, only 2K went on to the disambiguation page, so the primary topic seems plausible. -- JHunterJ (talk) 19:07, 10 January 2011 (UTC)

  It's probably because of the many links from other pages— Kallikanzarid^talk 20:30, 10 January 2011 (UTC)

  It's possibly because of that. I'm not sure how you can determine it's probably because of that. -- JHunterJ (talk) 20:33, 10 January 2011 (UTC)

  There're just so many of them :) — Kallikanzarid^talk 20:36, 10 January 2011 (UTC)

• Support. I can see no good reason why this is the primary topic: I would have thought the word used in the sense of 'machine operator' was much more common, and operator (programming) seems a more common usage than this too. So yes, and move the DAB page to Operator.--JohnBlackburne^words_deeds 00:53, 10 January 2011 (UTC)
• Comment: I agree that this move should happen, but it is clearly wrong, and silly, to say this meaning is unlikely to be sought by anyone searching for "operator". I see this happen all the time: people forget that the word "field" has different meanings in different.....fields, and link to field when they intend on particular one of them. People wanting to know about mathematics more often go straight to Wikipedia than do people wanting to know about other things; Wikipedia has been more successful in mathematics than in other fields. Michael Hardy (talk) 23:25, 10 January 2011 (UTC)
• Support, based on the many erroneous links. The current Operator (disambiguation) should be made the main page on operators. Jim.belk (talk) 18:25, 11 January 2011 (UTC)
• Comment: Please note that the new article name is still an issue; see User:Pohl's comment here, for example. — Kallikanzarid^talk 21:23, 11 January 2011 (UTC)
• Support. There is probably no primary topic. Operators in the sense of mathematics are certainly not the primary topic. Hans Adler 21:27, 11 January 2011 (UTC)
• Comment: The article begins by saying this:

An operator is a mapping from one vector space or module to another.

Normally I'd say that's not an appropriate opening sentence for a Wikipedia article, and I might edit it to say:

In mathematics, an operator is a mapping from one vector space or module to another.

However, if this were moved to operator (mathematics), then I'd say that initial context-setting phrase would be redundant. Michael Hardy (talk) 23:47, 11 January 2011 (UTC)

OTOH, I'd say that the current intro sentence is appropriate for a Wikipedia article. And, even if it's not appropriate, the appropriateness wouldn't change with any title change. -- JHunterJ (talk) 12:31, 12 January 2011 (UTC)

The problem as it stands is that the lay reader should find out in the first sentence that mathematics is what the article is about. The current first sentence certainly communicates that to mathematicians, but that might not be understood by laypersons. But obviously if it were moved to operator (mathematics) then the lay reader would know that without it's appearing in the first sentence. Michael Hardy (talk) 05:05, 13 January 2011 (UTC)

When I wrote the intro, the article was actually called operator (mathematics). I think it's OK to clarify the intro for the time being. — Kallikanzarid^talk 06:53, 13 January 2011 (UTC)

The {{about}} hatnote handles that clarification. Again, the lede phrasing isn't dependent on the current title. -- JHunterJ (talk) 12:19, 13 January 2011 (UTC)

Would it be OK if hatnote said This article is about operators on vector spaces and the article began with "In mathematics..."? — Kallikanzarid^talk 13:40, 13 January 2011 (UTC)

Can I suggest "This article is about mathematical operators ..." in the hatnote, and/or "An operator is a mathematical mapping from ..." in the lede? -- JHunterJ (talk) 13:47, 13 January 2011 (UTC)

The problem is that operators on vector spaces and algebraic operations are both sometimes called operators, so the two need to be disambiguated more diligently. — Kallikanzarid^talk 14:14, 13 January 2011 (UTC)

(mathematics) may be a lesser choice for a disambiguator then. This move might be to (vector mathematics) and Operator (algebra) might redirect to Operation (mathematics), or something. Operator (mathematics) could then be deleted or redirect to the disambiguation page (as an {{R from incomplete disambiguation}}). In the unlikely event that this article remains at the base name, {{two other uses|operators in vector spaces|algebraic operators|Operation (mathematics)}} yields:

This article is about operators in vector mathematics. For algebraic operators, see Operation (mathematics). For other uses, see Operator (mathematics) (disambiguation).

The lede already notes vector space, so that appears to be clear enough, with or without the insertion of "mathematical mapping". --JHunterJ (talk) 14:55, 13 January 2011 (UTC)

I like it, although the exact titles should be slightly different :)

• The article on algebraic operations already exists, it's called operation (mathematics). There is also an article operator algebra which is related to operators on vector spaces, not algebraic operations. And vector spaces as such are an algebraic topic. So I don't know if making operator (algebra) a redirect to operation (mathematics) will lessen the confusion, but it might.
• There is no 'vector mathematics', operators on vector spaces are studied mostly in linear algebra and functional analysis, the former focusing entirely on linear operators. I don't know what will be the best title for this article then. Maybe it's operator on a vector space, I don't know. — Kallikanzarid^talk 16:07, 13 January 2011 (UTC)

The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Deleted hatnote

''This article is about operators on vector spaces and modules. For applications of the concept to physics, see [[operator (physics)]], for algebraic operations, see [[operation (mathematics)]], for operators in programming languages, see [[operator (programming)]].''

I think the deletion was premature, because there is a confusion of terms, and a widely linked to article needs to briefly deal with it. — Kallikanzarid^talk 12:23, 13 January 2011 (UTC)

The confused terms are in the disambiguation page still in the hatnote. If there is some reason to call out those in addition, they should be combined in one hatnote. Hatnotes go above the issues templates, and are kept brief in general. -- JHunterJ (talk) 12:39, 13 January 2011 (UTC)

Linear operator: One vector space?

In linear algebra, I thought the term "linear operator" was reserved for linear transformations whose domain and range were the same vector space. For example,

Advanced Linear Algebra by Steven Roman, Third Edition, p. 59:

"A linear transformation from V to V is called a linear operator on V. The set of all linear operators on V is denoted by L(V). A linear operator on a real vector space is called a real operator and a linear operator on a complex vector space is called a complex operator." — Preceding unsigned comment added by 108.28.118.247 (talk) 15:49, 9 September 2011 (UTC)

operator of type (p,q)

What is an operator of type (p,q)? Citing from Jones/Kaufman/Rosenblatt/Wierdl: Oscillation in ergodic theory: "In any dynamical system the operator O defined in [formula number] (1.3) is weak type (1,1), type (p,p) for 1 < p < 1 and maps L1 to BMO." (MR1645330). --141.35.13.183 (talk) 15:02, 12 October 2011 (UTC)

References?

Actually, here in 2014, since many editors have now worked to separate programming from math, the article is taking shape IMO. However, I was shocked to see there are virtually no references! In that sense it "kinda is" a stub. I began by adding one on vectors, but will use Conway's wonderful operator book as well as several other journal articles and books to fill out the cites as a start, but certainly would appreciate help. I'm glad no one has deleted this yet for lack of citations as it is a noteworthy topic, and is looking much better than when computing was mixed in, but we need to begin filling in cites or one of the site's "hawks" will blow it away. I've also been working on a list and article on Polynomial Operators, and if you all agree, could do a section on it here too. There are now databases of CAS operators as the interface between math (function operators) and computing (operators in place of functions), but the distinction gets blurred in functional vs. imperative programming. My teaching area is CAS and numeric methods in computing, as well as robotics. Pdecalculus (talk) 17:37, 3 January 2014 (UTC)

Object and structure of the article

Copied from Wikipedia talk:WikiProject Mathematics:

This article was a mess. In particular, differential operators (among other important kinds of operators) are not even mentioned in the article. IMO, with the recent edits by an IP user, this article becomes worse. Other opinions are thus needed. D.Lazard (talk) 18:08, 19 March 2017 (UTC)

There is a problem that operator (mathematics) formerly discussed only linear operators, but linear operator already redirects to linear map. This was clearly not an ideal situation. What is a better configuration of topics? Sławomir Biały (talk) 18:25, 19 March 2017 (UTC)

As this concerns only a single article, it is better to continue the discussion on this article's talk page. I'll thus copy the above posts on Talk:Operator (mathematics), and reply there. D.Lazard (talk) 18:48, 19 March 2017 (UTC)

End of the copied text.

I suggest the following for the lead.

In mathematics, an operator is generally a mapping that acts on the elements of a space to produce other elements of the same space. The most common operators are linear maps, which act on vector spaces. However, when using "linear operator" instead of "linear map", mathematicians mean often actions on vector spaces of functions, which preserve also other properties, such that continuity. For example, differentiation and integration are linear operators, and operators that are built from them are called differential operators, integral operators or integro-differential operators.

Clearly, such a lead requires to rewrite the whole article. D.Lazard (talk) 19:16, 19 March 2017 (UTC)

Operator only from a space to itself?

The article starts with defining an operator as a function from a space to itself, but immediately goes on to the first section linear operators from one vector space to another. Linear operators are mostly not restricted to endomorphisms of vector spaces, but can be any linear function from a vector space U to a Vector space V (the Bronshtein defines it this way). So if you want linear operators to be operators in the general sense (which you don't have to), operators cannot be restricted to have the same codomain and domain. I will change that soon unless there are objections. — Preceding unsigned comment added by Ninjamin (talk • contribs) 11:36, 23 May 2019 (UTC)

Volterra Operator is not the indefinite Integration Operator (section Examples#Calculus)

For these two to be the same, you would have to restrict your functions to continuous functions and also to functions on intervals only or something like that. Calculus however does not only consider those very special functions only, so it shouldn't be called study of those two operators. In any case it's not good to consider differentiation and indefinite integration only, as an important part of calculus takes place where the definite integral cannot be based on an antiderivative. The Volterra operator is a special case of a definite integral, so I think it would be ok to just replace 'indefinite integral' with 'definite integral'. However, when I tried to replace 'indefinite integral' with 'definite integral', my edit was reverted, so I ask for other opinions. — Preceding unsigned comment added by Ninjamin (talk • contribs) 11:50, 23 May 2019 (UTC)

operator as symbol of an operation

Regarding the second meaning of "operator", the intro calls all operation symbols operators. So a symbol like the word "times" in "a times b" or the symbol "f" in "f(x)" is called an operator? It makes sense, I just wasn't familiar with this usage of the word. Can someone confirm that this is a common usage of this term? 109.40.2.142 (talk) 13:26, 4 June 2019 (UTC)

D. Lazard told me that in this context, operator as "symbol for an operation" can mean not only a special glyph for an operation, but also any other way to denote the operation, like changing typeface or using superscripts. Can we have some citation that those are also called operators in mathematics? In didn't find anything about that in the current citations. Actually, the citations using "operator" for "symbol of an operation" are written for/by teachers, computer scientists etc. Do mathematicians use "operator" for "symbol of an operation" at all? Ninjamin (talk) 10:04, 18 June 2019 (UTC)

WP:MULTI, my bad. D. Lazard answered this question here. Ninjamin (talk) 16:41, 14 July 2019 (UTC)

It is not correct to uniquely associate the word "Operator" within mathematics with Linear Operators

This article asserts that the term "Operator" within mathematics is associated only with Linear Operators.

I have the impression that the 'main curators' of this article (I get the impression there are such people though I may be incorrect) have a programme to purge all descriptions of Operators in a mathematical context other than as Linear Operators. I believe this may be carried out in good faith, as they believe other usages are either incorrect deviations, or incorrectly transferred from other contexts. However, from my experience this does not represent mathematical usage of the term generally (ie. it is a prescriptive rather than descriptive programme). Of course, my experience is also anecdotal (although it is supported by usage in numerous occasions throughout Wikipedia).

Examples of this programme are the separating out of general mathematical usage of the terms Operator and Operation into the article Operation (mathematics) only, the assertion that the term "Operator" in relation to arithmetic or logical mappings relates only to their notational symbols and not the concepts themselves, and that any direct association that is present between the word "Operation" and mappings is restricted to computer science only.

My personal view is that Operator in its most general usage simply means "special mapping" (both the concept and the symbol representing it depending on context).

This wiki informs about 2 distinct things with respect to mathematics; the logical concepts themselves, and the mathematical language/jargon and culture of the discipline. Both of these are empirical phenomena, useful to readers, and worthy of evidence. I believe the second aspect is especially important for foundational articles like this one, which act as landmarks for further exploration of mathematics.

I do not assert there are not potentially large or important communities within mathematics where the situation in the article is strictly true. Perhaps for example amongst experts in Linear Operators, or even the majority of all professional mathematicians (though personally I suspect not this widely). As important as these groups are, even they do not hold a monopoly on the language and culture of mathematics. I also do not assert the overloading and merging of these terms and concepts is beneficial. However, I do not also feel that the current article and the attempt it represents to direct the usage of these terms is useful either. Personally I feel a more direct approach to the current situation, variety of usage, and any potential confusion, would be more useful to people genuinely coming to this article to understand how "Operator" is used in mathematics.

I feel the firm distinction between mappings and the symbols that represent them is notably uncharacteristic of real mathematical usage. For example I believe most professional mathematicians would consider the "Levi-Civita symbol" really refers to an abstract concept rather than its notational representation (which is typically nothing more interesting than a regular epsilon).

I humbly ask other contributors not to delete this section, but rather to build up a library of evidence for and against this assertion. Of course my view is that eventually there will be a preponderance of non Linear Operator related usages. However I also welcome a demonstration that this is not the case.

Examples (on this wiki) where "Operator" is used to describe a mapping other than a linear mapping:

• Closure operator as of 3rd April 2020
• Operand as of 3rd April 2020 (there may be controversy with this assessment, but given that Operator and Operand are used
• Algebra of sets as of 3rd April 2020
• Boolean algebra

"Whitespace is used to specify logical AND, as it is the default operator for joining search terms"

• Logical connective is also known (as stated in the article) as Logical Operator or Sentential Operator
• Material conditional as of 4th April 2020 described in its article as a Logical Operator
• Negation as of 4th April 2020 described in some places within the article as an operator
• Group with operators — Preceding unsigned comment added by 82.69.43.178 (talk) 02:04, 24 April 2020 (UTC)

Examples outside this wiki

• I. J. Maddox - Elements of Functional Analysis - Page 6

Function

A function f is defined to be a relation, such that if (x,y)∈f and (x,z)∈f then y=z. Four other terms for function are map, mapping, operator and transformation.

• https://www.thefreedictionary.com/Mathematical+operator

2 Mathematics A function, especially one from a set to itself, such as differentiation of a differentiable function or rotation of a vector.

3 A logical operator.

• https://www.techopedia.com/definition/25582/arithmetic-operator
• https://math.stackexchange.com/questions/168378/what-is-an-operator-in-mathematics

3rd April 2020 - Shows a mix of views, roughly half in line with the current text of this article and half indicating a more general and less formal usage — Preceding unsigned comment added by 82.69.43.178 (talk) 05:40, 3 April 2020 (UTC)

• https://en.wiktionary.org/wiki/operator

(mathematics) A function or other mapping that carries variables defined on a domain into another variable or set of variables in a defined range. — Preceding unsigned comment added by 82.69.43.178 (talk) 05:44, 3 April 2020 (UTC)

• https://en.wiktionary.org/wiki/arithmetic_operator#English

(arithmetic) Any of the four basic operators in arithmetic, namely addition, subtraction, multiplication and division. — Preceding unsigned comment added by 82.69.43.178 (talk) 05:46, 3 April 2020 (UTC)

• https://www.merriam-webster.com/dictionary/operator

something and especially a symbol that denotes or performs a mathematical or logical operation

a mathematical function — Preceding unsigned comment added by 82.69.43.178 (talk) 05:49, 3 April 2020 (UTC)

• https://www.quora.com/In-mathematics-what-is-the-difference-between-an-operator-and-a-function?share=1

(as of 3rd April 2020) A mixture of responses, several indicating a more general and imprecise usage, one strongly supporting the current usage of the article

Notably:

Essentially, there is no difference. Operator is a function. The difference is mostly linguistical/contextual. In most cases operators are functions on vectors or functions (or functions considered as vectors).

So when I read about an operator, I expect that the input is a vector. When I read about a function, I expect the input to be a number.

- Dmitry Beliaev, Professor of Mathematics at University of Oxford — Preceding unsigned comment added by 82.69.43.178 (talk) 05:56, 3 April 2020 (UTC)

This has been already discussed here, for example in § Object and structure of the article. I have edited the lead of the article for, hopefully, making clear that operators are not necessary linear. It remain that the body of the article remains a mess. At least, some sections on nonlinear operators must be added. I am not ready for that, and hope that someone (maybe you) will do it. D.Lazard (talk) 10:31, 3 April 2020 (UTC)

Dear Lazard, thank you for your comment. I would love to open for suggestions / feedback on this topic to ensure the result is a consensus. It seems like there are several interested people. I am not very familiar with the Wikipedia editing process and do not know the best way to go about that. Particularly since I think this page should be considered in the context of other pages. I don't know if there is any feature provided in Wikipedia or discussing a 'topic' or group of articles?

I believe that given the closeness in the terms "Operator", "Operation" and "Symbol", (especially the distinction between actual symbols and concepts like Levi-Civita symbol or Legendre symbol) means it would be more effective to consider these articles collectively, rather than a constant gradual rearranging between the pages independently. Given the very close overlap between the concepts in Mathematics, Physics, and Computer Science I wonder if a similar consideration is worthwhile across these. Especially the association with Endomorphism and the issue of special notation, and surely history of the terms (though I know not much about that) which is common across these disciplines. To me it would perhaps make sense to unify these pages given the amount of overlap. Although of course, they each have their own important considerations, and they may also have separate pages. I wonder if it would make sense to have some overarching page such as "Operator (Mathematical sciences)" (though this exact title seems not great to me). I would prefer to get feedback from contributors across all these related articles if there was some way.

I wonder if perhaps and overview / summary page titled "Operator, Operation and Symbol" or similar would be useful?