Generator of a $C_0$-semigroup restricted to a subspace - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T17:30:27Z http://mathoverflow.net/feeds/question/117767 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/117767/generator-of-a-c-0-semigroup-restricted-to-a-subspace Generator of a $C_0$-semigroup restricted to a subspace Ollie Margetts 2013-01-01T05:41:43Z 2013-01-01T11:00:16Z <p>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 $$X_s P_{s,0} X_t V_{s+t,t} = X_{s+t}.$$</p> <p>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.</p> <p><strong>Question 1</strong>: Have such families of operators received some attention before? In particular I'm interested in properties of the "genrator" $d$. </p> <p><strong>Example</strong>: 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 $df=-f'$.</p> <p><strong>Question 2</strong>: Have such restrictions been studied in detail?</p> http://mathoverflow.net/questions/117767/generator-of-a-c-0-semigroup-restricted-to-a-subspace/117775#117775 Answer by András Bátkai for Generator of a $C_0$-semigroup restricted to a subspace András Bátkai 2013-01-01T10:32:09Z 2013-01-01T11:00:16Z <p>I do not think such semigroups have been <em>extensively</em> studied.</p> <p>I have seen such semigroups (and, more generally, evolution families corresponding to the non-autonomous problem) in</p> <p>A. Lunardi, M. Geissert, <a href="http://www.math.unipr.it/~lunardi/Files/OUasympt.pdf" rel="nofollow">Asymptotic behavior and hypercontractivity in nonautonomous Ornstein-Uhlenbeck equations</a>. J. Lond. Math. Soc. (2) 79 (2009), no. 1, 85--106.</p>