Let $\Omega$ be an open set in $\mathbb{R}^n$, $n\geq 1$.
If $H$ and $Z$ are Hilbert spaces and $T:\Omega \times H \to Z$ is an operator, suppose that for all $\omega \in \Omega_s \subset \Omega$ and all $h \in H$, we have $$T(\omega, h) = 0.$$
If $f:\Omega_s \to H$ is a map, so $f(\omega) \in H$ for all such $\omega$, am I allowed to deduce $$T(\omega, f(\omega)) = 0$$ with any assumptions on $T$ or $f$?