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.

  • 1
    $\begingroup$ Note that if $x_n$ converges weakly to $x$ then $$\|x\|\leq\liminf_{n\to\infty}\|x_n\|$$ $\endgroup$ Jun 1, 2019 at 11:24
  • $\begingroup$ @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. $\endgroup$
    – pitariver
    Jun 1, 2019 at 14:33

1 Answer 1


This result is a theorem (with proof) in Engel's book.

  • $\begingroup$ Spectacular book! Thanks for the reference. $\endgroup$
    – pitariver
    Jun 8, 2019 at 8:49
  • $\begingroup$ @pitariver The extended version is also great. $\endgroup$
    – Pedro
    Jun 8, 2019 at 23:57

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