Weighted Frobenius-Perron and Koopman operators
The author introduces operators \({\mathcal P}^u_\varphi: L_1 \to L_1\), \({\mathcal P}^u_\varphi= P_\varphi M_u\) and \(M_uU_\varphi: L_\infty \to L_\infty\), where \(P_\varphi\) and \(U_\varphi\) are the classical Frobenius-Perron and Koopman operators, respectively, and \(M_u\) is the corresponding multiplication operator. In other words, \({\mathcal P}^u_\varphi\) is defined by means of the equation \(\int_A {\mathcal P}^u_\varphi f \,d\mu = \int_{\varphi^{-1}(A)} u f \,d\mu\), \(A \in \Sigma\), and \(U_\varphi\) is defined by \(U_\varphi f = f \circ\varphi\). The author studies conditions under which \({\mathcal P}^u_\varphi\) and \(M_uU_\varphi\) are well-defined, bounded, have closed range, or are compact.
scientific article; zbMATH DE number 5716973