A Remark on the Norm of Integer Order Favard Spaces

摘要

For a generator $A$ of a $C_0$-semigroup $T(cdot)$ on a Banach space $X$ we consider the semi-norm $M^{k}_x:=limsup_{tto 0+}|t^{-1}(T(t)-I)A^{k-1}x|$ 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.