the infinitesimal generator
If a C_0 semigroup is the whole movie of an evolution — the family of operators that flow you forward by any amount of time — its infinitesimal generator is the velocity of that movie at the very first instant. It is the derivative of T(t) at t = 0, the single operator that encodes the instantaneous law of motion. Knowing the generator is knowing the differential equation; knowing the semigroup is knowing its solution flow. The generator is the compact thing, the semigroup is the full unfolding.
Precisely: given a C_0 semigroup {T(t)} on X, its generator A is defined by A x = limit as t goes to 0+ of (T(t)x - x)/t, with domain D(A) the set of x for which this limit exists. That quotient is literally a difference quotient in time, so A is 'd/dt at t = 0'. For the abstract Cauchy problem we usually write the equation as du/dt = -A u, so the generator of the heat semigroup is the (negative) Laplacian on its domain. The generator is typically unbounded and D(A) is only a dense subspace, not all of X — those are exactly the smooth-enough vectors. The fundamental link is that for x in D(A), the curve u(t) = T(t)x is differentiable and satisfies du/dt = A u (or -A u with the sign convention above): the semigroup solves the equation whose right-hand side is its generator.
This is the bridge between an abstract semigroup and a concrete PDE. You hardly ever build a semigroup by hand; instead you start from an operator A (an elliptic operator from a PDE), check the hypotheses of a generation theorem, and conclude that A generates a semigroup — which then is the solution operator you wanted. The generator's spectrum and resolvent (where (A - lambda I) is invertible) control everything downstream: existence via Hille-Yosida, decay rates via the spectral gap, smoothing via sectoriality.
For the heat semigroup T(t) = e^{t Laplacian} on L^2(R), take x = f smooth with compact support. Then (T(t)f - f)/t goes to Laplacian f = f'' as t goes to 0+, so the generator is A = Laplacian, with domain the f whose second weak derivative is again in L^2 (the Sobolev space H^2). The semigroup remembers the operator Laplacian as its first-instant velocity.
The generator is the time-derivative of the flow at t = 0 — it IS the differential operator.
A generator is closed and densely defined but almost never bounded, so it is NOT defined on all of X — feeding it a non-smooth vector is meaningless. A common error is to manipulate A x for x outside D(A); the domain is part of the operator, and getting it right (which boundary conditions, how much smoothness) is where the real content sits.