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_{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.