Undergraduate here. Some time ago I had a discussion with a friend on whether or not an operator is a function. The debate centered on this issue: I believed all operators are functions, while my friend believed not all operators are functions. I acknowledge the past questions asked regarding this issue, but intend for this question to build on previous discussions.
Argument: All operators are functions
My argument that an operator is a function is this: since the input and output of an operator is a function, then it fits the definition of a function:
"A relation $f$ is a function provided $(a,b),(a,c)\in f$ implies $b=c$."
In addition, the majority of online resources claim that operators are functions: On Quora, answers state that the operator is either "a special type of function" or exactly a function. In StackExchange, here, and here, the operator is also "a special kind of function," etc. Wikipedia states that an operator is a mapping, and since mappings are functions, operators are therefore functions. (I'll note that the reliability of Wikipedia is debated and questioned here.)
On my conceptual interpretation of the operator and the literature surrounding it, I came to support the argument that all operators are functions.
Argument: Not all operators are functions
My friend argued that there exist operators that do not conform to the definition of the function. He lists these operators as examples:
- The indefinite integral operator $\int$, in which $\int dt(f(t))=F(t)+C$ an arbitrary constant $C$.
- A square root operator $S$, where $S(f)=\pm\sqrt f$.
- A less-than operator $T$, where $T(f)=g$ for all $g<f$.
- A random number generator $U$, where $U(f)=x$ for some random number $x$.
Since for any of these functions, the input of a function can correspond to an output of two or more distinct functions, it follows that not all operators are functions. We agreed that if all operators are functions, then the indefinite integral operator and co. are not operators.
However, I also couldn’t ignore the literature online. I consulted with two professors. One suggested a correction on the indefinite integral, and it return an equivalence class of functions, instead of multiple distinct functions, for some function input, so technically there is one output. Although this argument fits the integral operator as a function, it seems to contradict the definition given by Wikipedia, where domain and codomain are the same. The other said that the definition of an operator depends on the math field we are working in.
Our (frankly enjoyable) debate thus stopped without conclusion, but I believed that progress would be made if we were given a rigorous (and preferably, widely accepted) definition of an operator, instead of working with what we think what an operator is. Thank you for reading, and my key questions are below:
- How are operators defined? If definitions vary depending on field, please give a working definition in your field.
- Are operators functions? Depending on the definition, this answer may differ. If operators are not functions, please give some counterexample.
(Some extra context: we both were in Intro to Quantum and read Gasiorowicz, and where I was first exposed to the idea of operators and operator manipulations such as $(\frac{d}{dx})^2=\frac{d^2}{dx^2}$. I thought if I really understand what an operator is, then this type of manipulation would be intuitive and founded. I also understand that there exists a relation between operators and transformations.)