I have been studying derivatives as operators on a function. Specifically, how we may write, for a function $f(x)$
$$\frac{df}{dx}$$
as
$$Df$$
where $D$ denotes $\frac{d}{dx}$. I've seen how successive application of an operator is seen as multiplying the operators together to give rise to a new operator. For example
$$\frac{d^2f}{dx^2}$$
can be written as $D^2f$.
What I have been pondering is this: Consider a certain operator $\Phi$, which has the property that
$$\Phi^2f=Df$$
i.e the $\Phi$ operator acts like a sort of "square root" of the derivative operator. This would imply that
$$\Phi f = h$$
and
$$\Phi h = g$$
where $g(x)$ and $h(x)$ are functions and that
$$\frac{df}{dx}=g$$
My question is, has such an operator been studied before? Does it have a unique name? I do not necessarily belive $\Phi$ to be unique, perhaps many unique operators may fill the role of $\Phi$. But can we determine if such an operator is unique or not?
