A Remark on the Norm of Integer Order Favard Spaces

For a generator $A$ of a $C_0$-semigroup $T(\cdot)$ on a Banach space $X$ we consider the semi-norm $M^{k}_x:=\limsup_{t\to 0+}\|t^{-1}(T(t)-I)A^{k-1}x\|$ on the Favard space ${\cal F}_{k}$ of order $k$ associated with $A$. The use of this semi-norm is motivated by the functional analytic treatment of time-discretization methods of linear evolution equations. We show that sharp inequalities for bounded linear operators on ${\cal D}(A^k)$ can be extended to the larger space ${\cal F}_{k}$ by using the semi-norm $M^{k}_{(\cdot)}$. We also show that $M^{k}_{(\cdot)}$ is a norm equivalent to the norms that are usually considered in the literature if A has a bounded inverse.