I would like to ask whether "evolution semigroups" are really useful (to discover something that can't be discovered in some other way?). There is a huge machinery to deal with them, but from my point of view the theory fails to provide new insight.
Let me recall the notion with a simple example. Say that you consider a differential equation $x'=A(t)x$, where $A(t)$ is an $n\times n$ matrix varying continuously with $t\in\mathbb R$. It induces what is sometimes called an evolution family $X(t,s)$ such that $X(t,s)x(s)=x(t)$ for any solution $x$.
The evolution semigroup (perhaps it would be better: evolution semiflow) associated to the equation is defined by $$(T(t)u)(s)=X(s,s-t)u(s-t)$$ for each continuous function $u\colon\mathbb R\to\mathbb R^n$. There are many other possibilities (of equations, spaces, etc), this is just an example.
So my question is:
What can we gain easier from looking at $T(t)$ instead of looking at $X(t,s)$?
As far as I understand the two have the same information (thus why the word "easier"), although the machinery developed is so huge, that I wonder whether I am missing something.
All becomes much trickier when we consider unbounded linear operators with dense domains that can in general depend on $t$.
Most of the theory that is now taught actually grew out abstract semigroups, starting with the work of Oliver Heaviside in signal analysis. The semigroups didn't come later; they came first. Heaviside reasoned very abstractly that if you start with the state of a fixed linear circuit at $t=0$, say $\phi_0$, and you evolve $t$ seconds into the future to obtain a state $E(t)\phi_0$, and then you evolve that new state $t'$ more seconds into the future, you would have $E(t')E(t)\phi_0$, which would have to be the same as starting with $\phi_0$ and evolving $t+t'$ seconds. That is, $E(t')E(t)=E(t'+t)$ would have to hold. This exponential property of time evolution is very general for systems that are not time-dependent. It's a property of time evolution.
From his evolution operator method, Heaviside derived operator methods for solving equations that Mathematicians had been unable to solve previously. The formalism that Heaviside defined gave the Laplace transform and its inversion integral. The Laplace transform is a tool that grew directly out of Heavside's methods. Heaviside was such a jerk that Mathematicians deliberately tried to keep his name out of everything. The Laplace transform was named after Laplace only because they found an integral resembling it that Laplace had once used. The Laplace method of solution is unquestionably Heaviside's. Heaviside invented the delta function as the derivative of the step function, as an impulse at time $t=0$; but Mathematicians chose to mock him instead by naming the step function after him. J.D. Jackson has shown how the Dirac delta was probably lifted by Dirac from Heaviside's work.
Heaviside created all kinds of operator methods based on these ideas, including the $D$ operator method taught in ODEs, the annihilator method, the cover-up method for partial fraction solutions associated with this method. None of these inventions carry Heaviside's name. Heaviside was the first to vectorize Maxwell's equations; the modern div, grad and curl operators are mostly his. Heavside is considered the Father of modern Electrical Engineering, and the foundations he laid were created in the most abstract way starting with these evolution methods.
When semigroups methods were studied in the context of Functional Analysis decades later, the Laplace transform was the natural tool for studying this subject. Heavside was taking square roots and applying other functions to operators before Functional Analysis was a subject.
One of my favorite quotes from an old Heavside paper follows, where he mocks Mathematicians who were demanding him to be rigorous when he was getting verifiable answers that they could not (you'll see why the man was hated):