In this paper, we extend the notion of diskcyclicity and disk transitivity of a single operator to a subset of $\mathcal{B}(X)$. We establish a diskcyclicity criterion and we give the relationship between this criterion and the diskcyclicity. As applications, we study the diskcyclicty of $C_0$-semigroups and $C$-regularized groups of operators. We show that a diskcyclic $C_0$-semigroup exists on a complex topological vector space $X$ if and only if dim$(X)=1$ or dim$(X)=\infty$ and we prove that diskcyclicity and disk transitivity of a $C_0$-semigroups and $C$-regularized groups are equivalent.

中文翻译：

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算子集和应用程序的磁盘循环性

在本文中，我们将单个运算符的磁盘循环性和磁盘传递性的概念扩展到 $\mathcal{B}(X)$ 的子集。我们建立了一个磁盘循环性标准，并给出了该标准与磁盘循环性之间的关系。作为应用程序，我们研究了 $C_0$-semigroups 和 $C$-regularized 运算符组的磁盘循环。我们证明了一个磁盘循环 $C_0$-半群存在于一个复杂的拓扑向量空间 $X$ 当且仅当 dim$(X)=1$ 或 dim$(X)=\infty$ 并且我们证明了磁盘循环性和磁盘传递性$C_0$-semigroups 和 $C$-regularized Groups 是等价的。