The (Weakly) Mixing Property of Frequently Hypercyclic Semigroups

Xiao-mei MO; Yong-lu SHU

doi:10.13718/j.cnki.xdzk.2019.08.009

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For a single operator, frequently hypercyclic operators are weakly mixing, and operators satisfying the frequent hypercyclcity criterion are topologically mixing. Based on the results in the researches of single frequently hypercyclic operators, we use the research methods of weak mixing and mixing to make a comparative analysis of the related properties of frequently hypercyclic operators with frequently hypercyclic semigroups, with the focus of discussion placed on the properties of frequently hypercyclic C₀-semigroups. First, we generalize the Erdös-Sárközy theorem to real number sets, and give a sufficient condition for judging positive real number sets to be syndetic sets, that is, if a known positive real number set has positive lower density, then its difference set is syndetic. Next, we prove that any frequent hypercyclic C₀-semigroup is weak mixing. Finally, we give a sufficient condition for C₀-semigroupsto be mixing. With the method of functional analysis, we prove that C₀-semigroups satisfying the frequent hypercyclcity criterion are mixing.

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设X为无限维可分的Banach空间，U，V⊂X为任意非空开集.如果存在n≥0，使得Tⁿ(U)∩V≠∅，则称算子T是拓扑传递的.如果T⊕T是拓扑传递的，则称T是弱混合的.如果存在N≥0，使得对∀n≥N，有Tⁿ(U)∩V≠∅，则称T是混合的.设$A \subset \mathbb{N}_{+}$，如果序列(n_k)_k⊂A单增，且$\sup\limits_{k \geqslant 1}(n_{k+1}-n_{k} )<\infty$，则称A为syndetic集. A为syndetic集等价于$\mathbb{N}_{+} \backslash A$中的连续整数区间的长度是有限的.类似地，$M \subset \mathbb{R}_{+}$是syndetic集等价于M的补集$\mathbb{R}_{+} \backslash M$包含的区间长度是有限的.文献[1]提出了频繁超循环算子这一概念.设$A \subset \mathbb{N}_{+}$，A的下密度被定义为：

如果存在x∈X(频繁超循环向量)，使得dens(N(x，U))＞0，则称T为频繁超循环的.关于它的更多研究结果详见文献[2-4].文献[5]首次提出了C₀-半群的超循环性.如果X上的算子族(T_t)_t≥0满足：T₀=I；对∀s，t＞0有T_t+s=T_sT_t；对∀x∈X，t≥0，有$\lim\limits_{s \rightarrow t} T_{s} x=T_{t} x$，则称(T_t)_t≥0为C₀-半群.如果存在t≥0，使得T_t(U)∩V≠∅，则称(T_t)_t≥0是拓扑传递的.如果(T_t⊕T_t)_t≥0是拓扑传递的，则称(T_t)_t≥0是弱混合的.如果存在t₀≥0，使得对∀t≥t₀，有T_t(U)∩V≠∅，则称(T_t)_t≥0是混合的.文献[6]将频繁超循环性引入到C₀-半群.设$M \subset \mathbb{R}_{+}$为可测集，M的下密度定义为

其中μ为$\mathbb{R}_{+}$上的Lebesgue测度.令(T_t)_t≥0为C₀-半群，如果存在x∈X(频繁超循环向量)，使得dens{R(x，U)}＞0，其中$R(x, U)=\left\{t \in \mathbb{R}_{+} : T_{t}(x) \in U\right\}$，则称(T_t)_t≥0为频繁超循环半群.学者们常常利用偏微分方程的解半群来探索PDE的本质.因此一些特定的PDE的解半群的性质研究得到了更多的青睐^[7-8]，在生物、物理、化学、工程等领域都有重要的应用^[9-11].

1. 弱混合半群的一个判定定理

文献[12-13]利用回复集刻画了算子的弱混合性.类似地，文献[14-15]证明了C₀-半群(T_t)_t≥0的弱混合性等价于对任意的非空开集U，V⊂X和0-邻域集W，有R(U，W)∩R(W，V)≠∅，其中R(U，W)={t≥0：T_t(U)∩W≠∅}.本文的定理1将Erdös-Sárközy定理推广到了正实数集合上.

定理1  设$M \subset \mathbb{R}_{+}$有正的下密度，则D=M-M={n-m：n，m∈M，n≥m}是syndetic集.

证  若D不是syndetic集，则存在$\left(n_{k}\right)_{k} \subset \mathbb{R}_{+}$，使得对$\forall k \in \mathbb{N} $都有$t_{1}+t_{2}+\cdots+t_{k} \in \mathbb{R}_{+} \backslash D$.因为dens(M)＞0，所以对∀m＞0，都有

令

则

对∀k≤m，存在N≥1，使得$\mu\left(M_{k} \cap[0, N]\right)>\frac{N+1}{m}$.设M_j∩M_k=∅(k=1，…，m)，从而

矛盾.因此，存在j＜k，使得M_j∩M_k≠∅，t_j+1+…+t_k∈D，这与t_k的选取矛盾，所以M-M是syndetic集.

利用定理1的结论，我们得到了频繁超循环C₀-半群的弱混合性质：

定理2  Fréchet空间X上的频繁超循环C₀-半群是弱混合的.

证  设W为0-邻域集，U，V⊂X为非空开集，(T_t)_t≥0为X上的频繁超循环C₀-半群，则存在非空开集U₀⊂U，使得T_t₀(U₀)⊂W.因为(T_t)_t≥0是频繁超循环的，则存在t₀≥0，使得T_t₀(U)∩W≠∅.设x是频繁超循环向量，则存在$M \subset \mathbb{R}_{+}$，dens(M)＞0，使得T_t(x)∈U₀(∀t∈M).又对∀t，s∈M，t＞s，有T_t₀+t-s(T_sx)=T_t₀(T_tx)∈W，从而

则dens(R(U，W))＞0，故R(U，W)是syndetic集.因为(T_t)_t≥0为C₀-半群，则T_-t(W)是0-邻域集，即对∀k≥0，存在0-邻域集W₀，使得T_t(W₀)⊂W(0≤t≤k).由拓扑传递性知，存在s＞k，y∈W₀，使得T_s(y)∈V，所以对∀0≤t≤k，有

即对∀k≥0，R(W，V)包含了长度为k的区间，所以R(U，W)∩R(W，V)≠∅，(T_t)_t≥0是弱混合的.

2. 拓扑混合半群的一个判定定理

可分的Fréchet空间上满足频繁超循环准则的算子是拓扑混合的^[3]，这一结论在C₀-半群上也成立.

定理3 设X为可分Fréchet空间，(T_t)_t≥0为C₀-半群，如果(T_t)_t≥0满足频繁超循环准则，即存在X₀⊂X，且X₀稠密，和一列映射S_t：X₀X(t＞0)，使得对∀x∈X₀，有

(ⅰ) T_tS_tx=x，T_tS_rx=S_r-tx(r＞t＞0)；

(ⅱ) $t \longmapsto T_{t} x \mathrm{d} t$和$t \longmapsto S_{t} x \mathrm{d} t$在[0，+∞)上是Pettis可积的.

则对∀x∈X₀，当t→+∞时，有T_tx→0，S_tx→0.

证对任意有界线性泛函φ∈X^*，∀x∈X₀，有φ(T_tx)→0.否则，存在ε₀＞0和$\left(t_{n}\right)_{n} \in \mathbb{R}_{+}$，使得

矛盾.

设X₀=(y_l)_l⊂X稠密且可数，则存在$\left\{N_{l}\right\}_{l \in \mathbb{N}} \subset \mathbb{N}$单增，对∀λ≤l和紧集K⊂[N_l，+∞)，有

令

则$\left\|T_{n+1} x-u_{l}\right\|<\frac{4}{2^{l}}$^[16].因(u_l)_l⊂X的稠密性，则对∀ε＞0，T_tx∈X，存在u_{l_i}，使得‖T_tx-u_{l_i}‖＜ε.从而存在u_l₁，u_l₂，当l₁，l₂充分大时，有

故存在N＞0，当s，t＞N时，有

因此(T_tx)_t依范数收敛.由极限的唯一性知T_tx→0(∀x∈X₀).

对∀x∈X₀，有S_rx∈X，X₀⊂X稠密，则对∀δ＞0，存在x_δ∈X₀，使得‖S_rx-x_δ‖＜δ.因T_r-t是连续的，故对∀ε₀＞0，存在δ₀(x_δ=x_δ₀)，当‖S_rx-x_δ₀‖＜δ₀时，有‖T_r-t(S_rx-x_δ₀)‖＜ε₀.易知‖T_r-tx_δ₀‖＜ε₀，于是

故S_tx→0(∀x∈X₀).

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The (Weakly) Mixing Property of Frequently Hypercyclic Semigroups

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