Our study is focused on the dynamics of weighted composition operators defined on a locally convex space \(E\hookrightarrow (C(X),\tau _p)\) with X being a topological Hausdorff space containing at least two different points and such that the evaluations \(\{\delta _x:\ x\in X\}\) are linearly independent in \(E'\). We prove, when X is compact and E is a Banach space containing a nowhere vanishing function, that a weighted composition operator \(C_{w,\varphi }\) is never weakly supercyclic on E. We also prove that if the symbol \(\varphi \) lies in the unit ball of \(A(\mathbb {D})\), then every weighted composition operator can never be \(\tau _p\)-supercyclic neither on \(C(\mathbb {D})\) nor on the disc algebra \(A(\mathbb {D})\). Finally, we obtain Ansari–Bourdon type results and conditions on the spectrum for arbitrary weakly supercyclic operators, and we provide necessary conditions for a composition operator to be weakly supercyclic on the space of holomorphic functions defined in non necessarily simply connected planar domains. As a consequence, we show that no composition operator can be weakly supercyclic neither on the space of holomorphic functions on the punctured disc nor in the punctured plane.

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连续函数空间上加权合成算子的超循环性

我们的研究集中于在局部凸空间\（E \ hookrightarrow（C（X），\ tau _p）\）上定义的加权合成算子的动力学，其中X是包含至少两个不同点的拓扑Hausdorff空间，使得评估\（\ {\ delta _x：\ x \ in X \} \）在\（E'\）中线性独立。我们证明，当X是紧致的并且E是包含无处消失函数的Banach空间时，加权合成算子\（C_ {w，\ varphi} \）永远不会在E上具有弱超循环性。我们还证明，如果符号\（\ varphi \）位于\（A（\ mathbb {D}）\）的单位球中，则每个加权合成算子在\（C（\ mathbb {D}）\）或圆盘代数\（A（\ mathbb {D}）\）上都不可能是\（\ tau _p \）-超循环。最后，我们获得了任意弱超循环算子在谱上的Ansari-Bourdon型结果和条件，并且为在非必然简单连接的平面域中定义的全纯函数空间上的复合算子提供了弱超循环的必要条件。结果，我们表明，无论是在穿孔圆盘上还是在穿孔平面上的全纯函数空间上，都没有成分算子可以是弱超循环的。