In the theory of Hille Yoshida, we have a (definitely real) banach space $L$ and on it a contractive, strongly continuous semigroup of operators. ($T_tT_s=T_{t+s}$ and $T_0=I$.) We then define the generator $L$ as the strong derivative, defined where it exists, of the semigroup at $0$. There is no possibility of a functional calculus relationship because we aren't even on a Hilbert space, but we still have that $e^{tL}=T_t$ in the sense that T_t is the strong limit of $(1-tL/n)^{-n}$ as $n\rightarrow \infty$.
Is there a relationship with how people use operator semigroups (such as $e^{it\Delta}$) in PDE? It seems to me like Hille Yoshida can't apply in the complex world, and luckily nor would it need to because we often have Hilbert spaces in PDE situations, and thus could assign meaning via the functional calculus instead. Whenever both assignments of meaning are possible, do they agree?
It seems reasonable to conjecture that $T_t$ is fully determined by $T_1$, at least in the Hille Yoshida setting. (I don't know much about how things are set up in PDE.) Is this true?
The semigroup solves Schrodinger or Heat types of equations $$ \frac{d}{dt} x(t) = Ax(t),\;\; x(0)=x_{0} $$ There is a functional calculus that looks like the Laplace transform: $$ \mathscr{L}_{A}\{f\} = \int_{0}^{\infty}f(t)T(t)\,dt. $$ So, for example, $$ \mathscr{L}_{A}\{f\star g\} = \mathscr{L}_{A}\{f\}\mathscr{L}_{A}\{g\}, $$ where $f\star g$ is the convolution of $f$ and $g$. This allows you to define powers of the generator under reasonable restrictions. And $$ \mathscr{L}_{A}\{e^{-st}\} = (sI-A)^{-1}. $$ If $f \in L^{1}[0,\infty)$, then the ordinary Laplace transform $\mathscr{L}\{f\}$ is holomorphic in the right half-plane, and you can view $\mathscr{L}_{A}\{f\}$ as $(\mathscr{L}\{f\})(A)$, which is related to the holomorphic functional calculus. I'm not being specific about conditions. You just asked about general ideas. These techniques work nicely for studying operators in Banach spaces, especially general $L^{p}$.