Hello unfortunately I can not show that the application "u" below is an endomorphism of E, if you can help me you are welcome, thank you.
These data :
It would help if you told us what 'E' was! Since you have u(f)= xf'(x), it must be at least 'differentiable functions' but are there any other conditions? And what operations are you allowing? That is, are you thinking of E as a group, vector space or algebra?