On systems theoretic aspects of Koopman operator theoretic frameworks

In this paper, we point out and study a connection between the recently flourishing consideration of Koopman operators and classical systems theoretic concepts such as aggregation and observability decompositions of nonlinear systems. The exploration of this newly unveiled cross-connection promotes a cross-fertilization of different methodologies and ideas intrinsic to the two different frameworks, resulting in a deeper understanding of both domains. In particular, the insights established in the paper connect intuitive systems theoretic viewpoints with the framework of Koopman operators.