2
$\begingroup$

In Pazy's book on semigroups he mentions (page 18) that when you have a commuting family of operators $B(t)$, such that $$ \sup \| B(t_1) .. B(t_n) \| \le M $$ for all finite choices $t_1, .. t_n$ then you can renorm the operators into a contraction, by choosing the sup of all such finite sets. So far so good. He further mentions that in general $$ \| B(t)^n \| \le M \qquad \forall t \forall n $$ will not be a sufficient condition in general whilst for resolvents (his case with Hille-Yosida) is does.

Regrettably, he does not give any example of hint. Do you have one, please?

$\endgroup$
2
$\begingroup$

If I understand your question correctly, then you will find your example in the paper

G. Nickel, R. Schnaubelt: An extension of Kato's stability condition for nonautonomous Cauchy problems. Taiwanese J. of Math. 2 (1998), 483-496.

Section 3 is of special interest, where counterexamples are constructed.

An online preprint version is available at this address.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.