Nemytskii Operators on Some Function Spaces

Abstract

AbstractFor a nonempty set S, Banach spaces X and Y over a field K = R or C, and a nonempty open set \(U\,\subset\,X\), let G, F, and H be vector spaces of functions acting from S into X,from U into Y, and from S into Y, respectively. Usually, but not always, F, G, and H will be normed spaces. Given a function \((u,s)\longmapsto,\psi(u,s)\) from U × S into Y, and a function g : \(S\,\rightarrow\,U\), the Nemytskii operator \(N\psi\) is defined by \(N\psi(g)(s)\quad:=\quad,N\psi\,g)(s)\quad:=\quad\psi(g(s),s)\qquad s\,\epsilon\,S.\) (6.1) Other authors call such an operator a superposition operator, e.g. Appell and Zabrejko [3].We use the term Nemytskii operator (as many others have) partly to distinguish it from the two-function composition operator \((F,G)\mapsto FoG\) to be treated in Chapter 8. Recall also that for a linear operator A we write Ax := A(x). We will often apply this rule also when A is a Nemytskii operator.KeywordsBanach SpaceBanach AlgebraMinkowski InequalityNonempty Open SubsetPoint PartitionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.