论文标题
两个超环体卷积操作员的家族
Two Families of Hypercyclic Non-Convolution Operators
论文作者
论文摘要
令$ h（\ mathbb {c}）$为所有功能的集合。令$λ，b \ in \ mathbb {c} $，令$ c_ {λ，b}：h（\ m马比布{c}）\ to h（\ mathbb {c}）$为构图操作符$ c_ {λ，λ，λ，b} f（z）f（z）= f（z）= f（λz+b）$，并且是$ d $ der $ der pertator peratian pertation pertation pertation pertation。我们扩展了非卷积操作员的超循环性$ t_ {λ，b} = c_ {λ，b} \ circ d $，通过显示$ | |λ| \ geq 1 $，operators \ okity \ begin \ begin {align*}时 \{ψ(T_{λ,b}): ψ(z)\in H(\mathbb{C}), ψ(0)=0 \text{ and } ψ(T_{λ,b}) \text{ is continuous}\} \end{align*} forms an algebra under the usual addition and multiplication of operators which consists entirely of超环算子（即，每个操作员都有一个密集的轨道）。我们还证明了操作员的集合\ begin {align*} \ {c_ {λ，b} \Circφ（d）：φ（z）\ text {是}φ（0）= 0 \} \ end End {Align*}的全部函数，完全由超循环运算符组成。
Let $H(\mathbb{C})$ be the set of all entire functions endowed with the topology of uniform convergence on compact sets. Let $λ,b\in\mathbb{C}$, let $C_{λ,b}:H(\mathbb{C})\to H(\mathbb{C})$ be the composition operator $C_{λ,b} f(z)=f(λz+b)$, and let $D$ be the derivative operator. We extend results on the hypercyclicity of the non-convolution operators $T_{λ,b}=C_{λ,b} \circ D$ by showing that whenever $|λ|\geq 1$, the collection of operators \begin{align*} \{ψ(T_{λ,b}): ψ(z)\in H(\mathbb{C}), ψ(0)=0 \text{ and } ψ(T_{λ,b}) \text{ is continuous}\} \end{align*} forms an algebra under the usual addition and multiplication of operators which consists entirely of hypercyclic operators (i.e., each operator has a dense orbit). We also show that the collection of operators \begin{align*} \{C_{λ,b}\circφ(D): φ(z) \text{ is an entire function of exponential type with } φ(0)=0\} \end{align*} consists entirely of hypercyclic operators.
