Let $A(t)$ be a family of skew self-adjoint operator defined on some Hilbert space $H$ with common domain $D(A).$ The dependence on $t$ is in the strongly continuous sense, i.e. for all $x \in D(A)$ the map $t \mapsto A(t)x$ is continuous.

Consider the initial value problem

$$\varphi'(t)=A(t)\varphi(t)$$

with $\varphi(0)=\varphi_0.$

Assume that there is a dense subspace $X$ in $H$ that is contained in the domain of $A(t)$ for all $t$ and $A(t)X \subset X.$

I ask:

Let $\varphi_0 \in X$. Does this imply that $\varphi(t) \in X$ for all $t>0?$

This sounds very natural but I do not have any tools/ideas to show this at the moment.