# Article

Full entry | PDF   (0.1 MB)
Keywords:
eventually differentiable semigroups; operator families
Summary:
In the first note we show for a strongly continuous family of operators \$(T(t))_{t\ge 0}\$ that if every orbit \$t\mapsto T(t)x\$ is differentiable for \$t>t_x\$, then all orbits are differentiable for \$t>t_0\$ with \$t_0\$ independent of \$x\$. In the second note we give an example of an eventually differentiable semigroup which is not differentiable on the same interval in the operator norm topology.
References:
[1] Bárta T.: On smooth solutions of Volterra equations via semigroups. Bull. Austral. Math. Soc. 78 (2008), 249–260. DOI 10.1017/S0004972708000683 | MR 2466862
[2] Batty C.J.K.: Differentiability of perturbed semigroups and delay semigroups. Perspectives in Operator Theory, 39–53, Banach Center Publ., 75, Polish Acad. Sci., Warsaw, 2007. MR 2336710 | Zbl 1126.47037
[3] Iley P.: Perturbations of differentiable semigroups. J. Evol. Equ. 7 (2007), no. 4, 765–781. DOI 10.1007/s00028-007-0349-0 | MR 2369679 | Zbl 1160.47037
[4] Pazy A.: Semigroups of Linear operators and Applications to Partial Differential Equations. Springer, Berlin, 1983. MR 0710486 | Zbl 0516.47023

Partner of