当前位置: X-MOL 学术Dokl. Math. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Symplectic Geometry of the Koopman Operator
Doklady Mathematics ( IF 0.6 ) Pub Date : 2021-09-22 , DOI: 10.1134/s1064562421040104
V. V. Kozlov 1, 2
Affiliation  

Abstract

We consider the Koopman operator generated by an invertible transformation of a space with a finite countably additive measure. If the square of this transformation is ergodic, then the orthogonal Koopman operator is a symplectic transformation on the real Hilbert space of square summable functions with zero mean. An infinite set of quadratic invariants of the Koopman operator is specified, which are pairwise in involution with respect to the corresponding symplectic structure. For transformations with a discrete spectrum and a Lebesgue spectrum, these quadratic invariants are functionally independent and form a complete involutive set, which suggests that the Koopman transform is completely integrable.



中文翻译:

Koopman 算子的辛几何

摘要

我们考虑由具有有限可数加性测度的空间的可逆变换生成的 Koopman 算子。如果这个变换的平方是遍历的,那么正交 Koopman 算子是一个在均值为零的平方可和函数的实 Hilbert 空间上的辛变换。指定了 Koopman 算子的无限二次不变量集,它们相对于相应的辛结构成对地对合。对于离散谱和勒贝格谱的变换,这些二次不变量在函数上是独立的,形成一个完整的对合集,这表明库普曼变换是完全可积的。

更新日期:2021-09-23
down
wechat
bug