Different definitions of $e^{t\Delta}$ All function spaces are over $\mathbb{R}^n$.  We have 3 different ways to define $e^{t\Delta}$:
(1)  For $f\in H^2$, we have $\widehat{\Delta f} (\xi)=-4\pi^2|\xi|^2\hat{f}(\xi)$.  Define $e^{t\Delta}$ as the map that satisfies $\widehat{e^{t\Delta} f} (\xi)=e^{-4\pi^2t|\xi|^2}\hat{f}(\xi)$.
(2) Define $e^{t\Delta}$ as the semigroup with infinitesimal generator $\Delta:H^2\subset L^2\rightarrow L^2$.
(3) Define $e^{t\Delta}$ by the appropriate functional calculus on the unbounded operator $\Delta$.
Are these equivalent definitions?  What are the advantages and disadvantages of each perspective?  I would guess (1) is easiest to define and get formulas from, and it's easier to generalize to $L^2(\Omega)$ with (2).  Are these in some sense ordered in increasing generality?  How are they interrelated?
 A: The three definitions are equivalent. (2) is a special case of (3), since unitary semigroups with self-adjoint generators are usually defined by functional calculus. The definition (1) is just passing from $\Delta$ to the unitary equivalent operator $\mathcal F^{-1}\circ\Delta\circ\mathcal F,$ where $\mathcal F: L^2\to L^2$ is the Fourier transform. Properties of the Fourier transform imply that $\mathcal F^{-1}\circ\Delta\circ\mathcal F$ acts as the multiplication operator 
$$f(\xi)\mapsto-4\pi^2|\xi|^2 f (\xi).$$ 
Exponential of this multiplication operator is just
$$f(\xi)\mapsto e^{-4\pi^2t|\xi|^2}f(\xi).$$
Since unitary equivalence interchanges with functional calculus, that is $$\varphi(\mathcal F^{-1}\circ\Delta\circ\mathcal F)=\mathcal F^{-1}\circ\varphi(\Delta)\circ\mathcal F,\ \varphi\in C(\mathbb R)$$ (1) and (3) are equivalent.
An advantage of (1) is that you easily derive the properties of $\Delta$ and $e^{t\Delta}$ from the properties of the multiplication operator, e.g. spectrum $\sigma(\Delta)=(-\infty,0]$ etc.
An advantage of (3) is that is easily seen to be well-defined.
Preferring (1) or (3) also depends on how do you define $\Delta.$ It is not trivial if you care about self-adjointness of the symmetric operator $\Delta.$
