On Ulyanov inequalities in Banach spaces and semigroups of linear operators
type: Item
instance of: scholarly article
title: On Ulyanov inequalities in Banach spaces and semigroups of linear operators
zbMATH Open document ID: 1190.47046
author: U. Westphal
author: Walter Trebels
published in: Journal of Approximation Theory
publication date: 2009-12-03T00:00:00Z
review text: An equibounded semigroup \(\{T_X(t): t\geq0\}\) of class \(C_0\) given on a Banach space \((X, \|\cdot\|_X)\) is supposed to be compatible with \(\{T_Y(t): t\geq0\}\) given on a Banach space \((Y, \|\cdot\|_Y)\). The fractional power \((-A)^\alpha\), \(\alpha>0\), of the semigroup generator is introduced according to the second author [\textit{U.\,Westphal}, Compos.\ Math.\ 22, 67--103 (1970; Zbl 0194.15401); in: Applications of fractional calculus in physics (World Scientific, Singapore), 131--170 (2000; Zbl 0989.44002)]: \[ (-A)^\alpha f=\lim_{\varepsilon\to 0} C_{\alpha, r}\int_\alpha^\infty u^{-\alpha,-1}(I-T(u)^r f \,du , \quad 0<\alpha<r, \] and an associated \(K\)-functional is defined as in [\textit{W.\,Trebels} and \textit{U.\,Westphal}, Constr.\ Approx.\ No.\ 3, 355--371 (2003; Zbl 1030.41021); Rend.\ Circ.\ Mat.\ Palermo (2) Suppl.\ 76, 603--620 (2005; Zbl 1144.47035)]. Under the assumption that \(\{T(t): t\geq0\}\) satisfies the Nikolskii type inequality with respect to compatible \(X\) and \(Y\): \(\|T(t)(T(t)f)\|_Y\leq c\varphi(t)\|T(t)f\|_X \) for all \(f\in X\) and \(\varphi\) such that \(\varphi(t)\geq\varphi(s)\), \(s>t>0,\) an abstract Ulyanov type inequality is derived between the (modified) \(K\)-functionals with respect to \((X, D_X((-A)^\alpha))\) and \((Y, D_Y((-A)^\alpha))\), where \(A\) is the generator of the semigroup \(\{T(t): t\geq0\}\). Known characterization of the \(K\)-functionals lead to certain Ulyanov type inequalities.
reviewed by: I. V. Melnikova
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 26A33
zbMATH DE Number: 5641729
zbMATH Keywords: Ulyanov inequality
zbMATH Keywords: Nikolskii inequality
zbMATH Keywords: \(K\)-functionals
zbMATH Keywords: semigroups of operators
zbMATH Keywords: fractional powers of generators
MaRDI profile type: Publication
full work available at URL: j.jat.2008.04.003
OpenAlex ID: W2020419573
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DOI: 10.1016/J.JAT.2008.04.003
description: scientific article; zbMATH DE number 5641729