A Landau-Kolmogorov inequality for generators of families of bounded operators
type: Item
instance of: scholarly article
title: A Landau-Kolmogorov inequality for generators of families of bounded operators
zbMATH Open document ID: 1205.47018
author: Carlos Lizama
author: Pedro J. Miana
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-09-01T00:00:00Z
review text: 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 \[ C_{a,k}:=\sup_{t>0}\frac{(a*a*k)(t)k(t)}{(k*a)^ 2 (t)} < \infty, \] and if \(A\) is a generator of an \((a,k)\)-regularized resolvent \(\{R(t)\}_{t\geq 0}\) such that \(\| R(t)\| \leq Mk(t)\), \(t\geq0\), with \(M\geq 1\), then the Landau-Kolmogorov inequality \[ \| Ax\|^ 2 \leq 8M^ 2 C_{a,k}\| x\| \| A^ 2 x\|, \] holds for all \(x\in D(A^ 2)\), where \(a\in L^ 1_{loc}(\mathbb R_ +)\) and \(k\in C(\mathbb R_ +)\) are positive functions. Examples are given to illustrate the obtained results.
reviewed by: James A. Oguntuase
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 5777335
zbMATH Keywords: Landau-Kolmogorov inequality
zbMATH Keywords: integrated semigroups
zbMATH Keywords: integrated cosine functions
zbMATH Keywords: abstract differential equations
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.05.065
OpenAlex ID: W2056670842
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DOI: 10.1016/J.JMAA.2010.05.065
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description: scientific article; zbMATH DE number 5777335