What is an operator in mathematics?


Based on your comment it sounds like you're actually asking about operations, not operators. A binary operation on a set $S$ is a special kind of function; namely, it is a function $S \times S \to S$. That is, it takes as input two elements of $S$ and returns another element of $S$. We can denote such an operation by a symbol such as $a \star b$ and then demand various additional properties of this operation, such as

  • associativity: $(a \star b) \star c = a \star (b \star c)$,
  • commutativity: $a \star b = b \star a$

and so forth. On the other hand, an arbitrary function $f : A \to B$ between two sets only takes a single input and returns an output which is not necessarily of the same type, so one can't speak of associativity or commutativity for such a thing. One might call a function $f : A \to A$ a unary operation but one still can't speak of associativity or commutativity for such a thing.

It is a pity that mathematics (being the most exact of all sciences) has some inexact (non-standardized) terms when it comes to certain cases. The terms operator and function are used as synonymous in certain texts but the term operator is used in a a narrower sense (as a special kind of function) in other texts. Hence, it is impossible to say one usage is correct and the other is incorrect. In my view

  1. The reader of a text has to use a concept according to and only according to the definition given in that particular text (contextual meaning),
  2. International standards has to fix these kinds of problems in the future.

Let $A$ and $B$ be ant two sets. Then $f:A\rightarrow B$ is said to be a $\textbf{function}$, if every element of $A$ is mapped to a unique element of $B$. Here requirement for $A$ and $B$ is only arbitrary sets.

Let $V_1,V_2$ be any two vector spaces. A map or a function $T:V_1 \rightarrow V_2$ is an $\textbf{operator}$. Here minimum requirement of $V_1,V_2$ be vector spaces. i.e., some algebraic structure should be there in domain and co-domain.