In our another recent article, we introduce a new dynamical property for linear operators called norm-unimodality which implies distributional chaos. In the present paper, we'll give a further discussion of norm-unimodality. It is showed that norm-unimodality is similar invariant and the spectra of norm-unimodal operator is referred to. As an application, in each nest algebra there exist distributional chaotic operators. Moreover, normal operators and compact operators with regard to norm-unimodality and Li-Yorke chaos are also be considered. Specially, a small compact perturbation of the unit operator could be distributionally chaotic.
Dynamical systems method (DSM) for unbounded operators
设$L$是实Hilbert空间中的无界线性算子，$H$是$C_0$半群的生成元，$g:H\to H$是$C^2_{loc}$非线性映射.求解方程$ $ F（v）的DSM（动力系统方法）：=Lv+gv=0 $ consists of solving the Cauchy problem $ \dot {u}=Φ（t，u）$, $ u（0）=u_0 $, where $ Φ $ is a suitable operator, and proving that i) $ \exists u（t）\quad \forall t>0 $, ii) $ \exists u（\infty）$, and iii) $ F（u（\infty））=0$.
Dynamical systems method for solving nonlinear equations with non-smooth monotone operators
考虑实希尔伯特空间中的算子方程（*）$B(u)+\ep u=0$，其中$\ep>0$是小常数。解方程（*）的动力系统方法（DSM）是一个柯西问题的构造，它具有以下性质：1）对于任意初始值，它有一个整体解; 2）当时间趋于无穷大时，这个解趋于一个极限; 3）这个极限解方程$B(u)=0$。利用DSM证明了方程（*）的唯一解的存在性，其中单调半连续算子$B$定义在所有的$ If $ \ep =0 $ and equation (**) $ B（u）=0 $ is solvable, the DSM yields$解上.
Metric on Nonlinear Dynamical Systems with Perron-Frobenius Operators
RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）；School of Fundamental Science and Technology, Keio University（庆应义塾大学基础理工学部）；The Institute of Scientific and Industrial Research, Osaka University（大阪大学产业科学研究所）
Typical dynamical properties of operators on $\ell_p$
Institute for Mathematical Sciences and Artificial Intelligence & Department of Mathematics, Shantou University（汕头大学数学与人工智能研究所及数学系）；Department of Mathematics, Shantou University（汕头大学数学系）
Dynamical System with Boundary Control Associated with Symmetric Semi-Bounded Operator
Let $L_0$ be a closed densely defined symmetric semi- bounded operator with nonzero defect indexes in a separable Hilbert space $\cal H$ . It determines a { \it Green system} $\{{\cal H}, {\cal B}; L_0, Γ_1, Γ_2\}$ , where ${\cal B}$ is a Hilbert space , and $Γ_i: {\cal H} \to \cal B$ are the operators related through the Green formula $$(L_0^*u, v)_{\cal H}-(u,L_0^*v)_{\cal H}=(Γ_1 u, Γ_2 v)_{\cal B} - (Γ_2 u, Γ_1 v)_{\cal B}.$$ The { \it boundary operators} $Γ_i$ are chosen canonically in the framework of the Vishik theory. With the Green system one associates a { \it dynamical system with b…
