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A Landau-Kolmogorov inequality for generators of families of bounded operators. (English) Zbl 1205.47018

A Landau-Kolmogorov type inequality for generators of a class of strongly continuous families of bounded linear operators on Banach space is investigated. In particular, the authors establish a Landau-Kolmogorov inequality when A is a generator of certain (a,k) regularized resolvents defined on a Banach space X. Specifically, they show that, if (a,k) is a pair satisfying the CP-condition and

C a,k :=sup t>0 (a*a*k)(t)k(t) (k*a) 2 (t)<,

and if A is a generator of an (a,k)-regularized resolvent {R(t)} t0 such that R(t)Mk(t), t0, with M1, then the Landau-Kolmogorov inequality

Ax 2 8M 2 C a,k xA 2 x,

holds for all xD(A 2 ), where aL loc 1 ( + ) and kC( + ) are positive functions. Examples are given to illustrate the obtained results.

MSC:
47A63Operator inequalities
26D10Inequalities involving derivatives, differential and integral operators
47A10Spectrum and resolvent of linear operators
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