Are there any other useful semigroups aside from $e^{At}$

Question

$e^{At}$ trivially satisfies both properties of semigroup namely

1. $T(t+s) = T(t)T(s)$
2. $T(0) = I$

Does there exist any other commonly used operators aside from $e^{At}$ that is a semigroup? Ignoring the obvious ones such as $M^{t}$ where $M \in \mathbb{R}$

Answer 1

The answer is yes. For examples, I suggest you the paper What is a semi-group?, by Einar Hille (page 55 of the book Studies in real and complex analysis edited by I. I. Hirschman, Jr.). In this paper, we read

One encounters one-parameter transformation semi-groups in a variety of problems of analysis. We shall give some instances.

1. Translations: The simplest of all such semi-groups is that of translations. We take $X = C[0, \infty]$ and define $$T(a)[f](u) = f(u + a),\quad a> 0$$ The semi-group property is immediate.

After, the author consider other examples (involving fractional integration, harmonic functions, stochastic processes, diffusion equations and ergodic theory).

Answer 2

More context? In certain situations, every one-parameter semigroup can be expressed in that way (e.g. every strongly-continuous one-parameter semigroup of bounded operators on a Banach space $X$ has an infinitesimal generator $A$, and thus can be expressed as $\exp(-At)$, where $A$ is a closed densely defined linear operator on $X$).