I am starting the study of semigroups of bounded linear operators, following the book by A. Lazy. He says the following in section 1.10: Let $T(t), t\ge 0,$ be a strongly continuous semigroup on a Banach space $X$. We can form the family of adjoints $T(t)^*$ of the operators $T(t)$, getting the semigroup $T(t)^*$ for $t\ge 0$. He claims that, however, $T(t)^*$ need not be a strongly continuous semigroup, because the map $T(t)\mapsto T(t)^*$ does not preserve strong continuity.
Well, I want to think of a counterexample but I don't have any intuition on how to create one. I know that I should choose a space $X$ which is not reflexive, as in the reflexive case we in fact have that the adjoint semigroup of a strongly continuous semigroup is strongly continuous. So maybe $L^1$ would be a good candidate.
Any suggestion on what semigroup to consider would be very much appreciated! Thank you very much!
Hint: translation is strongly continuous on $L^1(\mathbb R)$, but not on $L^\infty(\mathbb R)$.