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Keywords:
eventually differentiable semigroups; operator families
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:
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[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
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