On quasi-contractivity of C₀-semigroups on Banach spaces

Abstract.

A basic result in semigroup theory states that every C₀-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 not quasi-contractive.