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Zbl 1205.47018
Lizama, Carlos; Miana, Pedro J.
A Landau-Kolmogorov inequality for generators of families of bounded operators.
(English)
[J] J. Math. Anal. Appl. 371, No. 2, 614-623 (2010). ISSN 0022-247X

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 $$C\sb{a,k}:=\sup\sb{t>0}\frac{(a*a*k)(t)k(t)}{(k*a)\sp 2 (t)} < \infty,$$ and if $A$ is a generator of an $(a,k)$-regularized resolvent $\{R(t)\}\sb{t\ge 0}$ such that $\| R(t)\| \le Mk(t)$, $t\ge0$, with $M\ge 1$, then the Landau-Kolmogorov inequality $$\| Ax\|\sp 2 \le 8M\sp 2 C\sb{a,k}\| x\| \| A\sp 2 x\|,$$ holds for all $x\in D(A\sp 2)$, where $a\in L\sp 1\sb{loc}(\Bbb R\sb +)$ and $k\in C(\Bbb R\sb +)$ are positive functions. Examples are given to illustrate the obtained results.
MSC 2000:
*47A63 Operator inequalities, etc.
26D10 Inequalities involving derivatives, diff. and integral operators
47A10 Spectrum and resolvent of linear operators

Keywords: Landau-Kolmogorov inequality; integrated semigroups; integrated cosine functions; abstract differential equations

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