Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
continuous at zero semigroup. A.NIKNAM Ferdowsi university of Mashhad,Iran – user134885 Mar 12 '14 at 7:25
up vote 5 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.