西南大学学报 (自然科学版)  2019, Vol. 41 Issue (8): 54-57.  DOI: 10.13718/j.cnki.xdzk.2019.08.009
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  • 频繁超循环半群的(弱)混合性    [PDF全文]
    莫小梅, 舒永录     
    重庆大学 数学与统计学院, 重庆 401331
    摘要:对于单个算子而言,所有频繁超循环算子都是弱混合的,满足频繁超循环准则的算子都是拓扑混合的.在单个频繁超循环算子的研究成果的基础上,再结合单个算子弱混合和混合的研究方法,进一步对单个频繁超循环算子和频繁超循环半群的相关性质进行了对比分析,主要讨论了频繁超循环C0-半群的相关性质.首先,把Erd s-Sárk zy定理推广到了在实数集上,给出了判定正实数集合是syndetic集的一个充分条件,即已知一个正实数集合有正的下密度,则这个集合的差集是syndetic的.其次,证明了任意频繁超循环C0-半群是弱混合的.最后,给出了判定C0-半群是混合的一个充分条件.利用泛函分析的方法,证明了满足频繁超循环准则的C0-半群是混合的.
    关键词频繁超循环    C0-半群    弱混合    频繁超循环准则    混合    

    X为无限维可分的Banach空间,UVX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TT是拓扑传递的,则称T是弱混合的.如果存在N≥0,使得对∀nN,有Tn(U)∩V≠∅,则称T是混合的.设$A \subset \mathbb{N}_{+}$,如果序列(nk)kA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:

    $ \underline {{\mathop{\rm dens}\nolimits} } (A) = \mathop {\lim \inf }\limits_{N \to \infty } \frac{{{\mathop{\rm card}\nolimits} \{ n \le N:n \in A\} }}{N} $

    如果存在xX(频繁超循环向量),使得dens(N(xU))>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(Tt)t≥0满足:T0=I;对∀st>0有Tt+s=TsTt;对∀xXt≥0,有$\lim\limits_{s \rightarrow t} T_{s} x=T_{t} x$,则称(Tt)t≥0C0-半群.如果存在t≥0,使得Tt(U)∩V≠∅,则称(Tt)t≥0是拓扑传递的.如果(TtTt)t≥0是拓扑传递的,则称(Tt)t≥0是弱混合的.如果存在t0≥0,使得对∀tt0,有Tt(U)∩V≠∅,则称(Tt)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

    $ \underline {{\mathop{\rm dens}\nolimits} } (M) = \mathop {\lim \inf }\limits_{N \to \infty } \frac{{\mu (M \cap [0, N])}}{N} $

    其中μ$\mathbb{R}_{+}$上的Lebesgue测度.令(Tt)t≥0C0-半群,如果存在xX(频繁超循环向量),使得dens{R(xU)}>0,其中$R(x, U)=\left\{t \in \mathbb{R}_{+} : T_{t}(x) \in U\right\}$,则称(Tt)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(Tt)t≥0的弱混合性等价于对任意的非空开集UVX和0-邻域集W,有R(UW)∩R(WV)≠∅,其中R(UW)={t≥0:Tt(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-mnmMnm}是syndetic集.

      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

    $ \mathop {\lim \inf }\limits_{N \to \infty } \frac{{\mu (M \cap [0, N])}}{N} > \frac{1}{m} $

    $M_{k}=M+\left(t_{1}+t_{2}+\cdots+t_{k}\right) \quad k \in \mathbb{N}$

    $ \underline {{\mathop{\rm dens}\nolimits} } \left( {{M_k}} \right) = \underline {{\mathop{\rm dens}\nolimits} } (M) > \frac{1}{m} $

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

    $ \begin{array}{l} N \ge \mu \left( {\left( {{M_1} \cup {M_2} \cup \cdots \cup {M_m}} \right) \cap [0, N]} \right)\\ = \sum\limits_{k = 1}^m \mu \left( {{M_k} \cap [0, N]} \right) > N + 1 \end{array} $

    矛盾.因此,存在jk,使得MjMk≠∅,tj+1+…+tkD,这与tk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.

      设W为0-邻域集,UVX为非空开集,(Tt)t≥0为X上的频繁超循环C0-半群,则存在非空开集U0U,使得Tt0(U0)⊂W.因为(Tt)t≥0是频繁超循环的,则存在t0≥0,使得Tt0(U)∩W≠∅.设x是频繁超循环向量,则存在$M \subset \mathbb{R}_{+}$dens(M)>0,使得Tt(x)∈U0(∀tM).又对∀tsMts,有Tt0+t-s(Tsx)=Tt0(Ttx)∈W,从而

    $t_{0}+M-M \subset R\left(U_{0}, W\right) \subset R(U, W)$

    dens(R(UW))>0,故R(UW)是syndetic集.因为(Tt)t≥0C0-半群,则T-t(W)是0-邻域集,即对∀k≥0,存在0-邻域集W0,使得Tt(W0)⊂W(0≤tk).由拓扑传递性知,存在skyW0,使得Ts(y)∈V,所以对∀0≤tk,有

    $T_{s-t}\left(T_{t} y\right) \in T_{s-t}(W) \cap V$

    即对∀k≥0,R(WV)包含了长度为k的区间,所以R(UW)∩R(WV)≠∅,(Tt)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,(Tt)t≥0C0-半群,如果(Tt)t≥0满足频繁超循环准则,即存在X0X,且X0稠密,和一列映射StX0X(t>0),使得对∀xX0,有

    (ⅰ) TtStx=xTtSrx=Sr-tx(rt>0);

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

    则对∀xX0,当t→+∞时,有Ttx→0,Stx→0.

      对任意有界线性泛函φX*,∀xX0,有φ(Ttx)→0.否则,存在ε0>0和$\left(t_{n}\right)_{n} \in \mathbb{R}_{+}$,使得

    $\left|\varphi\left(T_{t_{n}} x\right)\right|>\varepsilon_{0} \quad \int_{0}^{+\infty}\left|\varphi\left(T_{t} x\right)\right| \mathrm{d} t>\sum\limits_{i=1}^{+\infty} \int_{t_{i}-\delta}^{t_{i}+\delta}\left|\varphi\left(T_{t_{i}} x\right)\right| \mathrm{d} t \geqslant+\infty$

    矛盾.

    X0=(yl)lX稠密且可数,则存在$\left\{N_{l}\right\}_{l \in \mathbb{N}} \subset \mathbb{N}$单增,对∀λl和紧集K⊂[Nl,+∞),有

    $\left\|\int_{K} T_{t} y_{\lambda} \mathrm{d} t\right\|<\frac{1}{l 2^{l}} \quad\left\|\int_{K} S_{t} y_{\lambda} \mathrm{d} t\right\|<\frac{1}{l 2^{l}}$

    $z_{n}=\left\{\begin{array}{ll}{y_{l}} & {n \in A\left(l, N_{l}\right)} \\ {0} & {其它}\end{array}\right.$
    $x=\sum\limits_{n \geqslant 1} \int_{n}^{n+1} S_{t} z_{n} \mathrm{d} t \quad u_{l}=\int_{0}^{1} T_{t} y_{l} \mathrm{d} t$

    $\left\|T_{n+1} x-u_{l}\right\|<\frac{4}{2^{l}}$[16].因(ul)lX的稠密性,则对∀ε>0,TtxX,存在uli,使得‖Ttx-uli‖<ε.从而存在ul1ul2,当l1l2充分大时,有

    $\left\|T_{t} x-u_{l_{1}}\right\|<\varepsilon \quad\left\|T_{s} x-u_{l_{2}}\right\|<\varepsilon$

    故存在N>0,当stN时,有

    $\begin{array}{l} \left\| {{T_s}x - {T_t}x} \right\| \le \left\| {{T_s}x - {u_{{t_1}}}} \right\| + \left\| {{u_{{t_1}}} - {T_{n + 1}}x} \right\| + \\ \left\| {{T_{n + 1}}x - {u_{{t_2}}}} \right\| + \left\| {{u_{{t_2}}} - {T_t}x} \right\| \le \varepsilon + \frac{4}{{{2^{{l_1}}}}} + \frac{4}{{{2^{{l_2}}}}} + \varepsilon \end{array}$

    因此(Ttx)t依范数收敛.由极限的唯一性知Ttx→0(∀xX0).

    对∀xX0,有SrxXX0X稠密,则对∀δ>0,存在xδX0,使得‖Srx-xδ‖<δ.因Tr-t是连续的,故对∀ε0>0,存在δ0(xδ=xδ0),当‖Srx-xδ0‖<δ0时,有‖Tr-t(Srx-xδ0)‖<ε0.易知‖Tr-txδ0‖<ε0,于是

    $\left\|S_{t} x\right\|=\left\|T_{r-t} S_{r} x\right\| \leqslant\left\|T_{r-t}\left(S_{r} x-x_{\delta_{0}}\right)\right\|+\left\|T_{r-t} x_{\delta_{0}}\right\|<2 \varepsilon_{0}$

    Stx→0(∀xX0).

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    The (Weakly) Mixing Property of Frequently Hypercyclic Semigroups
    MO Xiao-mei, SHU Yong-lu     
    School of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
    Abstract: 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.
    Key words: frequent hypercyclicity    C0-semigroup    weak mixing    frequent hypercyclicity criterion    mixing    
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