Boundedness of Koopman operators on Besov spaces and its applications
| Reference No. | 2022a012 |
|---|---|
| Type/Category | Grant for General Research-Short-term Joint Research |
| Title of Research Project | Boundedness of Koopman operators on Besov spaces and its applications |
| Principal Investigator | Isao Ishikawa（Center for Data Science, Ehime University・Associate professor） |
| Research Period |
May 23,2022. ～
May 27,2022. |
| Keyword(s) of Research Fields | Koopman operator, composition operator, Besov space |
| Abstract for Research Report |
The purpose of the proposed research is to clarify the relationship between the boundedness of Koopman operators on Besov spaces and dynamical systems on Euclidean spaces, which play a central role in the theory of partial differential equation and function space and can analyze smoothness and decay of functions. The Koopman operator on Besov space is a linear operator on Besov space determined by the operation of composing a dynamical system into functions from the right. In the 2000s, Mezic and other researchers found applications of this operator from the viewpoint of data analysis of physical phenomena, and in recent years, it has found applications in a wide range of scientific fields, not limited to physics. On the other hand, however, there are still many unsolved theoretical aspects. In particular, the boundedness of the Koopman operator is a fundamental property of the theoretical guarantee, but there are many unsolved aspects of this property as well. Thus we need to employ various knowledge in analysis in pure mathematics since the properties of the Koopman operator strongly depend on the properties of the considered function space. In this proposed research, we aim to completely determine a dynamical system when the Koopman operator defined in a higher-order Besov space is bounded through discussions in collaboration with Mr. Ikeda and Mr. Taniguchi, who are experts in the theory partial differential equation and function space, respectively. Furthermore, based on the theoretical results obtained here, we expect to find clues for new applications of Koopman operators in Besov spaces to data analysis through discussions with Yoshinobu Kawahara of Kyushu University, who is an expert in machine learning. |
| Organizing Committee Members (Workshop) Participants (Short-term Joint Usage) |
Masahiro Ikeda（RIKEN・Research scientist） Koichi Taniguchi（Tohoku University・Assitant professor） |