Accretivity and duality map in Banach space
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.
scientific article; zbMATH DE number 3964798