Is there a name for operators of the type $A: M \to M$

Question

In some theorem in functional analysis I have noticed that it is important to assume that an operator $A: M \to M$ where $M$ is some set plus conditions, as opposed to $A: M \to N, M \neq N$

Is there an official distinction between operators that maps an element back into the space (or into a space of the same dimension, etc.) versus operators that maps an element into a different set?

Answer 1

If $M$ is a vector space, then we refer $A$ as an endomorphism.

There are differences when a function takes an element from one space to an element in the same space.

Take for example:

$f:C[0,1]_{\|\cdot\|_1}\to C[0,1]_{\|\cdot\|_2}$, where

$\|\cdot\|_1 \to$ integral norm
$\|\cdot\|_2 \to$ maximum norm

Note that $C[0,1]$ is complete under maximum norm and incomplete under integral norm. So in this case, $f$ maps elements from an incomplete space to a complete space even though the spaces are same.