Theory of Koopman Operator toward Data Analysis

Abstract

The Koopman operator is defined as the pullback of a dynamical system on a function space. Koopman operators have been investigated as one of the most promising approaches for analyzing time series data generated by a nonlinear dynamical system. It is important to find data-driven methods to estimate the mathematical invariants of Koopman operators. Consequently, in this study, we explain the motivation and idea behind applying the Koopman operator theory to data analysis and introduce three topics pertinent to our recent progress on the theoretical aspect of Koopman operators with function space theory. We consider several types of function spaces in which Koopman operators act, for example, reproducing kernel Hilbert spaces and Besov spaces, and reveal the relationship between the boundedness of a Koopman operator and the behavior of the dynamical system. In addition, we explicitly compute the generalized spectrum of the Koopman operator of the one-sided full 2-shift.