Petrović, Srdan A note on composition operators. (English) Zbl 0673.47032 Mat. Vesn. 40, No. 2, 147-151 (1988). Let (X,\({\mathcal A},\mu)\) be a \(\sigma\)-finite measure space, and let \(\phi\) be a measurable transformation on X into X. If the linear mapping \(C_{\phi}f=f\circ \phi\) is bounded in \(L_ p(\mu)\) (1\(\leq p\leq \infty)\), then it is called the composition operator induced by \(\phi\). The main result of this paper is as follows: Theorem. Let (X,\({\mathcal A},\mu)\) be a non-atomic \(\sigma\)-finite measure space. Then no composition operator on \(L^ p(\mu)\) is compact, \(1\leq p\leq \infty\). Reviewer: V.S.Rabinovich Cited in 1 Document MSC: 47B38 Linear operators on function spaces (general) 47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators Keywords:composition operator PDFBibTeX XMLCite \textit{S. Petrović}, Mat. Vesn. 40, No. 2, 147--151 (1988; Zbl 0673.47032)