Talk:Operator (mathematics)

message: 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)
thread_title: Properties?

message: 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)
thread_title: confusing passage

message: I have a question of Physics about Math that is on
.
Why does
:\mathcal <P^2>={\Delta}P^2+<P>^2 ?
--HydrogenSu 09:48, 26 January 2006 (UTC)
replies: text: It's solved somewhere. Thanks for care-me boys.--HydrogenSu 11:41, 27 January 2006 (UTC)
thread_title:  A Question of Linear Operators 

message: 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)
thread_title:  Operators in calculus 

message: 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:
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)
thread_title:  Definition is too narrow 

message: 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)
replies: text: 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)
replies: text: 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)
replies: text: · 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)
replies: text: 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)

text: 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)
thread_title:  Notation of a symbolic operator 

message: 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)
replies: text: About function and mapping: my bad :)  —Preceding unsigned comment added by Kallikanzarid (talk • contribs) 13:15, 22 November 2010 (UTC)
thread_title:  I reworked the introduction 

message: 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)
replies: text: 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)
thread_title:  Split the article? 

message: 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)
thread_title:  I'm starting a revamp of the article 

message: 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)
thread_title:  Rename the article? 

message: 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. — Kallikanzaridtalk 12:23, 13 January 2011 (UTC)
replies: text: 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)
thread_title:  Deleted hatnote 

message: 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)
thread_title:  Linear operator: One vector space? 

message: 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)
thread_title:  operator of type (p,q) 

message: 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)
thread_title:  References? 

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Object and structure of the article 

message: 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)
thread_title:  Operator only from a space to itself? 

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

message: 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)
replies: text: 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)
replies: text: WP:MULTI, my bad. D. Lazard answered this question here. Ninjamin (talk) 16:41, 14 July 2019 (UTC)
thread_title:  operator as symbol of an operation