A Landau–Kolmogorov inequality for generators of families of bounded operators

A Landau–Kolmogorov type inequality for generators of a wide class of strongly continuous families of bounded and linear operators defined on a Banach space is shown. Our approach allows us to recover (in a unified way) known results about uniformly bounded C0C0-semigroups and cosine functions as well as to prove new results for other families of operators. In particular, if A is the generator of an α  -times integrated family of bounded and linear operators arising from the well-posedness of fractional differential equations of order β+1β+1 then, we prove that the inequality‖Ax‖2⩽8M2Γ(α+β+2)2Γ(α+1)Γ(α+2β+3)‖x‖‖A2x‖, holds for all x∈D(A2)x∈D(A2).