On quasi-contractivity of \(C_0\)-semigroups on Banach spaces (Q704961)

Let \(A\) generate a \(C_0\)-semigroup on a complex Banach space \(X\). Then (I) \(e^{tA}\) is quasi-contractive w.r.t. every equivalent norm on \(X\iff\) (II) \(A\) is bounded \(\iff\) (III) \(s-\lim\limits_{n\to\infty}(e^{{t\over n}A}P)^n\) exists for any bounded projection P and \(t\geq0\). For a Hilbert space version of these results, see the author's paper [J. Funct. Anal. 205, No.2, 401--413 (2003; Zbl 1054.47032)].