Essential norm of substitution operators on \(L^p\) spaces
Let \((X,\Sigma, \mu)\) be a \(\sigma\)-finite measure space. Suppose that \(\varphi:X \to X\) is a non-singular transformation, that is, \(\mu\circ \varphi^{-1} \ll \mu\), and let \(u\) be a \(\Sigma\)-measurable function on \(X\). By the Radon-Nikodým theorem, for every \(f\in L^1\) there exists a \(\mu\)-unique function \(\tilde{f}_{\varphi}^{u}\in L_{\text{loc}}^{1}\) such that \(\int_{\varphi^{-1}(A)}uf\,d\mu=\int_A \tilde{f}_{\varphi}^{u}\,d\mu\) for every \(A\in \Sigma\). The mapping \({\mathcal P}_{\varphi}^{u}:f\mapsto \tilde{f}_{\varphi}^{u}\) is the weighted Frobenius-Perron operator on \(L^1\) associated to \((u,\varphi)\). The adjoint of \({\mathcal P}_{\varphi}^{u}\) on \(L^\infty\) is the weighted composition operator \(uC_\varphi\). In the paper under review, lower and upper estimates for the essential norm of finite sums of weighted Frobenius-Perron operators on \(L^1\) and weighted composition operators on \(L^p\) are given under some additional conditions.
scientific article; zbMATH DE number 6112151