On quasi-contractivity of $C_0$-semigroups on Banach spaces

Details On quasi-contractivity of $C_0$-semigroups on Banach spaces On quasi-contractivity of $C_0$-semigroups on Banach spaces

2006-11-30

arxiv.org/abs/math/0611935v1

Arch. Math. (Basel) 83 (2004), no. 4, 360--363

Mathematics

Functional Analysis

4 pages

Scientific paper

A basic result in semigroup theory states that every $C_0$-semigroup is quasi-contractive with respect to some appropriately chosen equivalent norm. This paper contains a counterpart of this well-known fact. Namely, by examining the convergence of the Trotter-type formula $(e^{\frac{t}{n}A}P)^n$ (where $P$ denotes a bounded projection), we prove that whenever the generator $A$ is unbounded it is possible to introduce an equivalent norm on the space with respect to which the semigroup is {\it{not}} quasi-contractive.

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