Bounded $C_0$-semigroups on barrelled spaces are equicontinuous

Question

I have the following question: Let $X$ be a barrelled locally convex space (every absolutely convex, absorbing and closed set is a neighborhood of zero) and let $(T(t))_{t\geq0}$ be a $C_0$-semigroup, i.e., $T(t)x\to x$ if $t\to0$ for all $x\in X$, such that $(T(t))_{t\geq0}$ is bounded in the following sense: for all bounded subsets $B$ of $X$ and for all (continuous) seminorms $q$ there exists a $M\geq0$ such that $\sup_{x\in B}{q(T(t)x)}\leq M$.

The claim I want to prove is that this is equivalent to being an equicontinuous semigroup, i.e., for all seminorms $p$ there exists another seminorm $q$ and $C\geq0$ such that $p(T(t)x)\leq Cq(x)$ for all $t\geq0$ and all $x\in X$. In other words: A $C_0$-semigroup on a barrelled locally convex space is bounded if and only if $(T(t))_{t\geq0}$ is equicontinuous.

Another question that arose during this investigation is if the same holds true if one substitutes "barrelled" by "C-sequential" (and probably "semi-Montel")?

Thanks in advance for your constructive feedback, I am looking forward hearing from you soon.

Answer 1

The Banach-Steinhaus theorem holds for barrelled domain $X$ and arbitrary locally convex codomain $Y$. Therefeore, every family $\{T(t):t\in I\}$ of operators $T(t)\in L(X,Y)$ with $\sup\{q(T(t)x): t\in I\}<\infty$ for every $x\in X$ and every continuous seminorm $q$ on $Y$ is equicontinuous.