# Generator of a $C_0$-semigroup restricted to a subspace

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?

• 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 Jan 4, 2013 at 10:34