Accretivity and duality map in Banach space
type: Item
instance of: scholarly article
title: Accretivity and duality map in Banach space
zbMATH Open document ID: 0598.47060
author: Hidekazu Asakawa
published in: Proceedings of the Japan Academy. Series A
publication date: 1984-01-01T00:00:00Z
review text: A multi-valued operator A in X is said to be accretive if (M) for any [x,x'],[y,y']\(\in A\), there exists \(f\in F(x-y)\) such that (x'-y',f)\(\geq 0,\) or equivalently if (A) for any [x,x'],[y,y']\(\in A\) and \(t>0\), \(\| (x,y)+t(x'-y')\| \geq \| x-y\|,\) where [x,x']\(\in A\) means that \(x\in D(A)\) and x'\(\in Ax\). The following lemma, which was first proved by \textit{T. Kato} [J. Math. Soc. Jap. 19, 508-520 (1967; Zbl 0163.383)], implies the equivalence of the two definitions. Lemma. Let x and y be elements of X. Then (M)' there exists \(f\in F(x)\) such that (y,f)\(\geq 0\), if and only if (A)' \(\| x+ty\| \geq \| x\|\) for any \(t>0.\) The purpose of this note is to give a new proof of the ''if'' part of this lemma. We use only the Hahn-Banach-Alaoglu theorem.
Mathematics Subject Classification ID: 47H06
zbMATH DE Number: 3964798
zbMATH Keywords: multi-valued operator
zbMATH Keywords: accretive
MaRDI profile type: Publication
full work available at URL: pjaa.60.201
OpenAlex ID: W2048758672
cites work: On accretive sets in Banach spaces
cites work: Orthogonality and Linear Functionals in Normed Linear Spaces
cites work: Nonlinear semigroups and evolution equations
cites work: Q3905599
cites work: Q5736344
DOI: 10.3792/PJAA.60.201
description: scientific article; zbMATH DE number 3964798