  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.
