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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 $Dom(d)$ 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 $\eta\in Dom(d)$ 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 "genrator" $d$.

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?

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I do not think such semigroups have been extensively studied.

I have seen such semigroups (and, more generally, evolution families corresponding to the non-autonomous problem) in

A. Lunardi, M. Geissert, Asymptotic behavior and hypercontractivity in nonautonomous Ornstein-Uhlenbeck equations. J. Lond. Math. Soc. (2) 79 (2009), no. 1, 85--106.

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Thanks. I see the evolution operators $P_{s,t}$ are mapping between different spaces, but I think it will take me some time to really work out what they're doing and whether this relates to these compressed semigroups – Ollie Margetts Jan 4 '13 at 10:34
A similar discussion can be found in Pazy's book. It is usually expected that such semigroups appear in the study of nonautonomous problems, but it unclear whether some of them may restated in "usual" terms by making the problem autonomous. – John B Dec 20 '15 at 22:55

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