On continuous composition operators

Wilhelmina Smajdor¹

¹ Technical University of Technology Kaszubska 23 PL-44-100 Gliwice, Poland

Annales Polonici Mathematici, Tome 98 (2010) no. 3, pp. 273-282.

Résumé

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.

Détail

DOI : 10.4064/ap98-3-6

Keywords: subset bbb interval normed linear space banach space investigate banach space lip functions psi psi sup psi infty where psi frac s r psi t s psi t r psi t r t s s r psi lip only psi differentiable its derivative psi lipschitzian suppose composition operator generated times rightarrow varphi varphi maps set mathcal affine functions varphi rightarrow lip prove continuous psi leq constant where psi varphi quad continuous linear rightarrow lip lipschitzian lder composition operators investigated