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I'm thinking specifically of the derivative here in that $\frac{\mathrm{d}}{\mathrm{d}x}\left[e^x\right]=e^x$. I realize that the fact that $e^x$ is its own derivative is, for the derivative operator itself, unique to the derivative acting on $e^x$.

I was wondering if there is a term for a non-trivial operator (i.e., a function taking a function as its argument) that returns the original function.

I wasn't thinking that either $\frac{\mathrm{d}}{\mathrm{d}x}$ or $e^x$ were special by themselves so much as the pair $\left\{\frac{\mathrm{d}}{\mathrm{d}x},\,e^x\right\}$ was particularly good example of the relationship. I guess I was looking for a term for the converse relationship that a fixed point has to its function. That is, if $e^x$ is a fixed point of $\frac{\mathrm{d}}{\mathrm{d}x}$, $\frac{\mathrm{d}}{\mathrm{d}x}$ is a _________ with respect to $e^x$.

The identity function/operator is, for the purpose of the question, trivial becauseit always returns the value/function in its argument, and idempotence only refers to the function/operator composed with itself. I am interested in distinct function/ operators $f$ and $g$ such that $g[f]=f$ for some $f$s in the domain.

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    The identity. ${}{}$ – 2017-02-08
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    $f$ is called a fixed point of $g$, but in the other direction there's too much variation. – 2017-02-08
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    Identity? I mean, if $g$ leaves $f$ unaltered for all $f$ then it is the identity. But if it depends on $f$, then I don't know if it has its own name. – 2017-02-08
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    I think [fixed point](https://en.wikipedia.org/wiki/Fixed_point_(mathematics)) is the closed I know of – 2017-02-08
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    When there an element $g$ of a group acts trivially on *every* function, we say the action is 'not faithful.' For linear maps, we call vectors that are sent to scalar multiples of themselves 'eigenvectors.' Also possibly related is the notion of idempotence. A function is idempotent when $f(f(x)) = f(x)$. This would be true in your case also, but again, only for the special value. – 2017-02-08

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The general term for a point that is unchanged by some mapping (such as an operator or a function) is a fixed point. $e^x$ is a fixed point of $\frac{d}{dx}$.

The mapping for which all points are fixed points is called the identity and is denoted by $id$, $e$ or $1_X$ depending on the context.

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You're thinking of it the wrong way around. $d/dx$ isn't special because it preserves $e^x$, $e^x$ is special because it's preserved by $d/dx$.

In this sense, you want to look into fixed points or eigenvectors.

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    I wasn't thinking that either $\frac{\mathrm{d}}{\mathrm{d}x}$ or $e^x$ were special by themselves so much as the pair $\left\{\frac{\mathrm{d}}{\mathrm{d}x},\,e^{x}\right\} $ was special. I guess I was looming for a term for the converse relationship that a fixed point to its function. That is, if $e^x$ is a fixed point of $\frac{\mathrm{d}}{\mathrm{d}x}$, $\frac{\mathrm{d}}{\mathrm{d}x}$ is a _________ with respect to $e^x$. – 2017-02-08
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    Eigenvector might be up your alley then, in that (matrix, eigenvector) is special. – 2017-02-08