Let $x=(x_i)$ be a sequence in $\ell^1$ such that all $x_i>0.$

Let $T(t):\ell^1 \rightarrow \ell^1$ be a strongly continuous semigroup of, i.e. $t \mapsto T(t)y$ is continuous for every $y \in \ell^1$.

We can now study

$$F_x(t):=\left\Vert \frac{1}{\sqrt{x}} (T(t)x-x) \frac{1}{\sqrt{x}} \right\Vert_{\ell^{\infty}}.$$

Here $$\frac{1}{\sqrt{x}}:=\Big(\frac{1}{\sqrt{x_i}} \Big)_i.$$

In general, this will be infinite for $t \neq 0.$

I ask: Under what natural conditions on $x$ and $(T(t))$, or rather the generator of the group, is this finite in a neighbourhood $(-\varepsilon,\varepsilon)$ of $t=0$, for fixed $x$, and we will have $$\lim_{t \downarrow 0} F_x(t)=0.$$

Observations:

The only danger that can happen is that an entry of $\frac{1}{\sqrt{x_i}}$ will be very large and $T(t)x$ will have a lot of mass in this entry, i.e. $T(t)x_i$ is then large and $\frac{1}{\sqrt{x_i}}$ is large and if this happens infinitely often, with increasing size, we are doomed.

So is there a way to ensure by looking at the generator that arbitrarily small entries do not get filled too quickly? Of course I still want my dynamics to mix the entries, just not in the way I described before.