Consider an analytic semigroup $e^{At}$ on the Hilbert space $H$, and linear operator $G:L^2(0,T;H)\to C(0,T;H)$ as $$x(t)=Gu(t):=\int_0^te^{A(t-s)}u(s)ds$$ The question is
If the sequence $u_n$ converges weakly to an element $u$ in $L^2(0,T;H)$; does $x_n$ converge strongly to $x$ in $C(0,T;H)$?
Some notes:
From analyticity of $e^{At}$, it can be shown that $x_n$ belong to $C^{0,1/2}(0,T;H)$.
If needed, domain of $A$ can be assumed to be a compact subspace of $H$.