Defining an mathematical operator in Mathematica


I’m trying to learn mathematica to use in my thesis. I have basic problems with definitions, calculations etc. Let me ask a question.

I want to define anti derivative operator and derivative operator, then a new operator with composition of these operators. Compositions must be identity, so if I give a function into new operator it will turn out the function what I gave it.

To explain better, I will try to write these in mathematical sense;

Here $ D^m$ denotes standard derivative operator $ m$ times and anti-derivative operator $ I_m$ defined with

\begin{cases}  f(x), & m=0 \ \int_0^{x} \dfrac{(x-y)^{m-1}}{(m-1)!}f(u)du, &  m\geq 1 \end{cases} 

which seems

Editor gives an error so I use code option and screenshot to code

Now, if we composite these functions, $ [D^m\circ I_m](f)(x)$ these will give us directly $ f(x)$ . I want to define these operators in Mathematica and see composition of operators.

Actually these problem is not my main problem but the answers will shows me what I need, then I will try to learn myself again.

Thanks for your helps, if I wrote incomprehensible or wrong forgive me.