I have the following problem;

Fix a Hilbert space $\mathcal{H}$. Let $S \colon \mathrm{Dom}S \subset L_2(\mathbb{R}_+, \mathcal{H}) \rightarrow L_2(\mathbb{R}_+, \mathcal{H}) $ be a closed densely defined (possibly unbounded) linear operator. Do we know when can we find such a family of operators $(T_t)_{t \geq 0}$ on $\mathcal{H}$, such that $$(Sf)(s)= T_s(f(s)) $$ for all $f \in L_2(\mathbb{R}_+, \mathcal{H})$ and $s \geq 0$. One condition which I observed is to make $S$ of the form $S= I_{L_2(\mathbb{R}_+)} \otimes X$, where X is a closed densely defined linear operator on $\mathcal{H}$. But I was just wondering maybe there are some other special conditions which will guarantee the existence of a 'nice' family $(T_t)_{t \geq 0}$ which will satisfy the condition which I mentioned. Thank you for any help.