Suppose we have a decreasing filtration of Banach spaces $(E_t)_{t\geq0}$, inclusions $V_{s+t,t}:E_{s+t}\to E_t$ and projections $P_{t,s+t}:E_t\to E_{s+t}$ such that $P_{t,s+t}V_{s+t,t}=I_{E_{s+t}}$. I want to study maps $X_t:E_t\to E_0$ satisfying $X_0=I$, $X_tV_{t,0}^*\to I$ strongly as $t\to0$ and $$X_s P_{s,0} X_t V_{s+t,t} = X_{s+t}. $$
Define a "generator" $d:E_0\to E_0$ by letting $\operatorname{Dom}(d)$ consist of elements $\eta\in E_0$ such that $$\eta\in E_t \text{ for some } t\geq0 ~~\text{ and } ~~\lim_{s\to 0} s^{-1} (X_s\eta-V_{s,0}\eta) \text{ exists}, $$ and setting $d\eta=\lim_{s\to 0} s^{-1} (X_s\eta-V_{s,0}\eta)$. Some results of $C_0$-semigroup theory translate directly to results about these families, for instance $\eta\in \operatorname{Dom}(d)$ whenever there exists $\varepsilon>0$ such that the section $(s^{-1} (S_s\eta-V_{s,0}\eta))_{s\in(0,\varepsilon)}$ exists and is uniformly bounded.
Question 1: Have such families of operators received some attention before? In particular I'm interested in properties of the "generator" $d$.
Example: Let $S=(S_t)_{t\geq0}$ be the semigroup of right shifts on $L^2(\mathbb{R}_+)$. If we try to restrict $S$ to the subspace $L^2[0,1]$, it does not give a semigroup, but shatters into a family $(T_t,H_t)_{t\geq0}$, where $H_t=L^2[0,1-t]$ if $t\leq 1$ and $H_t=\{0\}$ if $t>1$ and $T_t=S_t|H_t\to H_0$. The family $(T_t,H_t)$ has precisely the structure given above. The generator $d$ turns out to be closable with closure the differential operator on $L^2[0,1]$, with domain consisting of absolutely continuous functions with $L^2$ derivative and boundary conditions $f(0)=f(1)=0$ and defined by $df=-f'$.
Question 2: Have such restrictions been studied in detail?
