Nemytskii Operators on Some Function Spaces

Abstract

For 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.