We prove that a finite set of natural numbers J satisfies that $J\cup \{0\}$ is not Sidon if and only if for any operator T, the disjoint hypercyclicity of $\{T^j:j\in J\}$ implies that T is weakly mixing. As an application we show the existence of a non-weakly mixing operator T such that $T\oplus T^2\oplus\cdots \oplus T^n$ is hypercyclic for every n.

中文翻译：

---

不相交超循环、Sidon 集和弱混合算子

我们证明有限自然数集J满足$J\cup \{0\}$不是 Sidon 当且仅当对于任何算子T ， $\{T^j:j\in J\的不相交超循环性}$意味着T是弱混合的。作为一个应用，我们证明了非弱混合运算符T的存在，使得$T\oplus T^2\oplus\cdots \oplus T^n$对于每个n都是超循环的。