# Renorming into contraction

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?

## 1 Answer

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.