Suppose I have a solution $u \in L^2(0,T;H^1(\Omega))$ with $u' \in L^2(0,T;H^{-1}(\Omega))$ of the PDE $$u' + Au = f$$ where $A:L^2(0,T;H^1(\Omega)) \to L^2(0,T;H^{-1}(\Omega))$ is an elliptic operator and $f \in L^2(0,T;L^2(\Omega))$.
If, for some reason, I knew that $u'$ lies in the stronger space $L^2(0,T;L^2(\Omega))$, is it valid to say that since $$Au = f-u' \in L^2(0,T;L^2(\Omega)),$$ that $Au \in L^2(0,T;L^2(\Omega))$?
I can't see anything wrong with this reasoning but it appears wrong that we have $Au$ in a stronger space, since the operator $A$ is given a priori...