Title:

Two notes on eventually differentiable families of operators (English) 
Author:

Bárta, Tomáš 
Language:

English 
Journal:

Commentationes Mathematicae Universitatis Carolinae 
ISSN:

00102628 (print) 
ISSN:

12137243 (online) 
Volume:

51 
Issue:

1 
Year:

2010 
Pages:

1924 
Summary lang:

English 
. 
Category:

math 
. 
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. (English) 
Keyword:

eventually differentiable semigroups 
Keyword:

operator families 
MSC:

47D06 
idZBL:

Zbl 1222.47066 
idMR:

MR2666077 
. 
Date available:

20100521T12:30:33Z 
Last updated:

20130922 
Stable URL:

http://hdl.handle.net/10338.dmlcz/140085 
. 
Reference:

[1] Bárta T.: On smooth solutions of Volterra equations via semigroups.Bull. Austral. Math. Soc. 78 (2008), 249–260. MR 2466862, 10.1017/S0004972708000683 
Reference:

[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. Zbl 1126.47037, MR 2336710 
Reference:

[3] Iley P.: Perturbations of differentiable semigroups.J. Evol. Equ. 7 (2007), no. 4, 765–781. Zbl 1160.47037, MR 2369679, 10.1007/s0002800703490 
Reference:

[4] Pazy A.: Semigroups of Linear operators and Applications to Partial Differential Equations.Springer, Berlin, 1983. Zbl 0516.47023, MR 0710486 
. 