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 \(\mathcal {L}^2([-1,1])\).

在本文中，我们给出了一个镜像结构，并获得了与此镜像结构相关的镜像傅立叶变换\（\ mathcal {F} _ {\ lambda} \）。随后，我们考虑非交换Volterra方程\（S _ {\ theta} V _ {[-1,0]} = e ^ {\ mathbf {i} \ theta} V _ {[-]] S _ {\ theta} \）使用此镜像傅立叶变换\（\ mathcal {F} _ {\ lambda} \），其中\（V _ {[-1,0]} \）是复希尔伯特空间\（\ mathcal { L} ^ 2（\ mathbf {1} _ {[-1,0]} \ mathbb {R}）\）。然后，我们获得\（\ Vert（1 + V）g（t）\ Vert \ ge \ Vert g（t）\ Vert \）和\（\ Vert（1 + V ^ {*}）g（t）\ Vert \ ge \ Vert g（t）\ Vert \），其中V是复数希尔伯特空间\（\ mathcal {L} ^ 2 [0,1] \）上的Volterra运算符。这样\（1 + V \），\（1 + V ^ {*} \），\（（1 + V）^ {-1} \）和\（（1 + V ^ {*}）^ { -1} \）既不是超循环的也不是Li–Yorke混沌的，其中\（\ theta \ in \ mathbb {R} \）和\（（S _ {\ theta}）_ {\ theta \ in \ mathbb {R}} \）是\（\ mathbb {R} \）上的可逆镜像运算符的函数。事实上，我们的结构是正确的斜对称的Volterra操作者小号上复Hilbert空间\（\ mathcal {L} ^ 2（[ - 1,1]）\） 。