Mirror Operator and Its Application on Chaos

Abstract

In this paper, we give a mirror construction and we obtain a mirror Fourier transform $\mathcal {F}_{\lambda }$ associated with this mirror construction. Subsequently, we consider the noncommutative Volterra equation $S_{\theta }V_{[-1,0]}=e^{\mathbf {i}\theta }V_{[-1,0]}S_{\theta }$ using this mirror Fourier transform $\mathcal {F}_{\lambda }$, where $V_{[-1,0]}$ is the Volterra operator on the complex Hilbert space $\mathcal {L}^2(\mathbf {1}_{[-1,0]}\mathbb {R})$. Then, we obtain $\Vert (1+V)g(t)\Vert \ge \Vert g(t)\Vert$ and $\Vert (1+V^{*})g(t)\Vert \ge \Vert g(t)\Vert$, where V is the Volterra operator on the complex Hilbert space $\mathcal {L}^2[0,1]$. Such that $1+V$, $1+V^{*}$, $(1+V)^{-1}$ and $(1+V^{*})^{-1}$ are neither hypercyclic nor Li–Yorke chaotic, where $\theta \in \mathbb {R}$ and $(S_{\theta })_{\theta \in \mathbb {R}}$ is a function of invertible mirror operator on $\mathbb {R}$. In fact, our construction is right for the skew-symmetric Volterra operator S on the complex Hilbert space $\mathcal {L}^2([-1,1])$.