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A note on composition operators. (English) Zbl 0673.47032

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

MSC:

47B38 Linear operators on function spaces (general)
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
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