Why is a strongly continuous one-parameter semigroup called a $C_0$-semigroup?

Question

Why is a strongly continuous one-parameter semigroup called a $C_0$-semigroup? Isn't $C_0$ the set of continuous functions that vanish at infinity?

Thanks and regards!

Answer 1

In the context of semigroups of operators, $C_0$ or $(C,0)$ abbreviates Cesàro summable of order zero.

This simply means the continuity property $\lim\limits_{t \to 0^+} T_t x = x$ for all $x \in X$ (convergence in norm) and has nothing to do with continuous functions of compact support or vanishing at infinity.

The notation is explained and discussed in detail in the classic treatise on semigroups: Hille and Phillips, Functional analysis and semi-groups. They impose this and other (weaker) conditions on semi-groups such as $C_1$, Cesàro summability of order one, meaning that $x = \lim\limits_{t \to 0^+} \frac{1}{t} \int_{0}^t T_s x\,ds$, or variants of Abel summation.

See Section 10.6, p.320f for a discussion of these and various other classes of semigroups.