On continuous composition operators
Volume 98 / 2010
Abstract
Let $I \subset \Bbb R$ be an interval, $Y$ be a normed linear space and $Z$ be a Banach space. We investigate the Banach space Lip$_{2}(I,Z)$ of all functions $\psi:I\to Z$ such that $$ M_{\psi}:=\sup \{\|[r,s,t;\psi]\|: r< s< t,\, r,s,t\in I\}<\infty, $$ where $$ [r,s,t;\psi]:=\frac{(s-r)\psi(t)+(t-s)\psi(r)-(t-r)\psi(s)} {(t-r)(t-s)(s-r)}. $$ We show that $\psi\in$ Lip$_{2}(I,Z)$ if and only if $\psi$ is differentiable and its derivative $\psi'$ is Lipschitzian. Suppose the composition operator $N$ generated by $h:I \times Y\rightarrow Z$, $$ (N\varphi)(t):= h(t,\varphi(t)), $$ maps the set $\mathcal{A}(I,Y)$ of all affine functions $\varphi: I\rightarrow Y$ into Lip$_{2}(I,Z)$. We prove that if $N$ is continuous and $M_{\psi} \leq M$ for some constant $M>0$, where $\psi(t)=N(t,\varphi(t))$, then $$ h(t,y)=a(y)+b(t), \quad\ t \in I, \,y \in Y, $$ for some continuous linear $a:Y\rightarrow Z$ and $b \in $ Lip$_{2}(I,Z)$. Lipschitzian and Hölder composition operators are also investigated.
