# A weakly continuous semigroup of operators on a Banach Space is Strongly Continuous

The following is an exercise from Dunford & Schwartz (1958) page 653.

Let $$X$$ be a Banach space and $$T_t:X \to X$$ be a semigroup of bounded operators indexes by $$\mathbb{R}_{\geq0}$$. Suppose it's continuous w.r.t the weak operator topology, i.e. $$\forall \; x \in X, \; y^* \in X^*, \; t \to y^*(T_t(x)) \; cont.$$ Prove that it's strongly continuous, i.e. $$t \to T_tx$$ is norm continuous for all $$x\in X$$.

What I Tried:

Obviously we can only prove continuity at $$t=0$$, and one can show using applications of the uniform boundness principle that $$\Vert T_t \Vert \leq Me^{ct}$$. And From here we can reuce to the case of a semigroup of contractions on a Banach Space. But In general, I am not sure that if $$x_n \overset{w}{\to} x$$ weakly and $$\underset{n\to \infty}{limsup}\Vert x_n \Vert \leq \Vert x \Vert$$ then there is norm convergence.

Any help is appreciated.

• Note that if $x_n$ converges weakly to $x$ then $$\|x\|\leq\liminf_{n\to\infty}\|x_n\|$$ Jun 1, 2019 at 11:24
• @uniquesolution I don't see how it helps us here, the limsup condition is the other direction, and even with it I don't see how to prove strong convergence. Jun 1, 2019 at 14:33