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 C 0 -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 + 1 then, we prove that the inequality ‖ A x ‖ 2 ⩽ 8 M 2 Γ ( α + β + 2 ) 2 Γ ( α + 1 ) Γ ( α + 2 β + 3 ) ‖ x ‖ ‖ A 2 x ‖ , holds for all x ∈ D ( A 2 ) .