We can view the weak derivative as an operator $\partial_t :H^1_0(0,T) \subseteq L^2(0,T) \to L^2(0,T)$, where we can for instance take $H^1_0(0,T)$ to mean the weakly differentiable functions which vanish at zero. However, in some contexts it may be natural to consider instead $\widetilde \partial_t : L^2(0,T) \subseteq H^{-1}(0,T) \to H^{-1}(0,T)$ given by
$$ \langle \widetilde \partial_t f, g \rangle_{H^{-1}\times H^1} := -\langle f, \partial_t g \rangle_{L^2}, \quad f \in L^2(0,T), g \in H^1_0(0,T). $$ Are the operators $\partial_t$ and $\widetilde \partial_t$ related in a nice way? For instance regarding their spectra. I tried to see if I could prove that $\widetilde \partial_t$ is closed, but I got stuck very quickly, so I was wondering whether there is a viewpoint which makes these types of questions easier to handle.