A Closed Range Theorem for the Frobenius-Perron Operator and Its Application to the Spectral Analysis

We use the theory of functional analysis to obtain some new general results of Frobenius-Perron operators Ps:L1(X)→L1(X) and Koopman operators US:L∞(X)→L∞(X). As an application of these results, we get new necessary and sufficient conditions for PS and US to be isometric, respectively, and so in this case the spectrum of PS is completely determined.