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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 $\left(a,k\right)$ regularized resolvents defined on a Banach space $X$. Specifically, they show that, if $\left(a,k\right)$ is a pair satisfying the $CP$-condition and

${C}_{a,k}:=\underset{t>0}{sup}\frac{\left(a*a*k\right)\left(t\right)k\left(t\right)}{{\left(k*a\right)}^{2}\left(t\right)}<\infty ,$

and if $A$ is a generator of an $\left(a,k\right)$-regularized resolvent ${\left\{R\left(t\right)\right\}}_{t\ge 0}$ such that $\parallel R\left(t\right)\parallel \le Mk\left(t\right)$, $t\ge 0$, with $M\ge 1$, then the Landau-Kolmogorov inequality

${\parallel Ax\parallel }^{2}\le 8{M}^{2}{C}_{a,k}\parallel x\parallel \parallel {A}^{2}x\parallel ,$

holds for all $x\in D\left({A}^{2}\right)$, where $a\in {L}_{loc}^{1}\left({ℝ}_{+}\right)$ and $k\in C\left({ℝ}_{+}\right)$ are positive functions. Examples are given to illustrate the obtained results.

##### MSC:
 47A63 Operator inequalities 26D10 Inequalities involving derivatives, differential and integral operators 47A10 Spectrum and resolvent of linear operators
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