Let $T(t)_{t \ge 0}$ be a strongly continuous semigroup on a Hilbert space $H.$

Then, one can consider the function

$f(t_1,t_2):= T(t_1)S T(t_2)x$ where $x$ is a fixed element of the Hilbert space and $S$ a bounded operator.

Obviously this function is continuous componentwise (by strong continuity of the semigroup). I am wondering however whether it is also continuous as a map $f:\mathbb R_{\ge 0}^2 \rightarrow H$

Let $T(t)_{t \ge 0}$ be a strongly continuous semigroup on a Hilbert space $H.$

Then, one can consider the function

$f(t_1,t_2):= T(t_1)S T(t_2)x$ where $x$ is a fixed element of the Hilbert space and $S$ a bounded operator.

Obviously this function is continuous componentwise (by strong continuity of the semigroup). I am wondering however whether it is also continuous as a map $f:\mathbb R_{\ge 0}^2 \rightarrow H.$

Obviously, continuity holds if the semigroups are uniformly continuous, so the question is only interesting in the case of only strongly continuous semigroups.