Crystal算子的Hypercyclicity

Hypercyclicality of crystal operators

称复可分Banach空间上的算子T是Crystal算子,如果T在其任意非零不变子空间上的限制相似于T。对于某个x∈X,称T是超循环的(Hypercyclic),若{x,Tx,T2x,…,Tnx,…}=X;若∨{Tnx:n≥0}=X,则称T是循环的;称T是严格循环的,如果存在x∈X使得A(T)x=X,其中A(T)是T生成的闭子代数。研究Crystal算子T的循环性:1.Crystal算子都是循环算子,但不是严格循环的;2.利用向量值解析函数的性质,证明:当1∈σ(T)\σe(T)时,T是超循环算子。

An operator T on the complex separable infinite dimensional Banach space X is said to be crystal if T TIM for each nonzero invariable subspaee M of T. Say that T is Hypercyclic if there exists x ∈ X, such that {z,Tx,T2x,…,Tnx,…}=x;V{Tx：n≥0}=X, then T is said to be cyclic. Say that T is strictly cyclic if there exists x ∈ X such that A （T） x = X, where A （T） is the smallest closed sub-algebra containing T. The cyclicity of crystal operator T is considered. It is proven that ： 1. Crystal operator T is cyclic, but isn＇ t strictly cyclic ; and 2. T is Hypercyclic if1∈σ（T）／σe（T）.