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 alpha-times integrated family of bounded and linear operators arising from the well-posedness of fractional differential equations of order beta+ 1 then, we prove that the inequality parallel to Ax parallel to(2) <= 8M(2) Gamma(alpha+beta+2)(2)/Gamma(alpha+1)Gamma(a + 2 beta + 3) parallel to x parallel to parallel to A(2)x parallel to holds for all x is an element of D(A(2)) (C) 2010 Elsevier Inc. All rights reserved