Understanding linear operators

I have some basic problem with understanding linear operators (at least
in the area of numerical analysis) and I will be a very happy person if
you can help me out.

Let's say that I have a certain operator. Let's say that it's a simple
D operator, i.e. D[U(x)] = dU/dx. This far it's OK, and I understand
what I see. I even understand what D^4(U) means - it means a 4th
derivative by x of U. However, when I have an expression
Exp[dt*D^2]U(x), I'm lost. What's that suppose to mean? How it will
look in the combination of different forms of U(x)?

Same goes for such a basic operator as a shift operator E(U). E(U) =
U(x+h). It's fine, it's easy. However, what does ln(E) means is not
clear to me.

Can anyone please enlighten me? This is frustating, cause it doesn't
seem like a very hard thing.

.