Bn上的Bloch型空间Βω和小Bloch型空间Βω,0分别定义为:
![](https://cimg.fx361.com/images/2023/0413/0a3c9cb30e7e4dfa699ab58aff68a8e5917ef0aa.webp)
在范数‖f‖Βω=|f(0)|+‖f‖ω下,容易验证Bloch型空间和小Bloch型空间都是Banach空间.对这个空间的研究可见文[1]、[2]等.进一步,若取ω(r)=(1-r2)α,分别取α=1和0<α<1,则Βω是Bloch空间和Lipschitz空间.对Bloch空间和Lipschitz空间的研究可见文[8]、[9]、[10]等.
给定正规函数μ(z)=μ(|z|),Bn上的Zygmund型空间定义为:
![](https://cimg.fx361.com/images/2023/0413/01a03b25dbd87ecbd30aa95deca945e1e3551c92.webp)
其中,μ(z)=μ(|z|).进一步,称f属于小Zygmund空间Ζμ,0,如果f∈Ζμ且满足
容易验证Ζμ和Ζμ,0在范数
![](https://cimg.fx361.com/images/2023/0413/2007ee54fdeadc44261edb38b890be84fef39dc4.webp)
![](https://cimg.fx361.com/images/2023/0413/96aa1c0fd024b2bb0fa58fe13502eeed45bac6ba.webp)
下是Banach空间.
设g∈H(Bn),H(Bn)上的广义Cesàro算子Tg定义为:
![](https://cimg.fx361.com/images/2023/0413/b4c5ca10ec4321745792662e61533c32a0056e79.webp)
φ=(φ1,…,φn)是Bn上的全纯自映射,复合算子Cφ定义为:
![](https://cimg.fx361.com/images/2023/0413/60a408d975ff5b00d75eaa202719b9820ddf2ec5.webp)
广义Cesàro算子和复合算子的积为:
通过计算有:
![](https://cimg.fx361.com/images/2023/0413/a2a7f57473d606ca511dc17885a045086685016a.webp)
![](https://cimg.fx361.com/images/2023/0413/d728477b2a71009de383f579f8023ec4321240ab.webp)
其中,ℜφ(z)=(ℜφ1(z),…,ℜφn(z)).以下所出现的φ和ℜφ(z)都如这里所述.若n=1,g=φ,φ(0)=0,该算子就是Volterra型复合算子;若φ(z)=z,该算子就是广义Cesàro算子(见文[4]、[6]、[7]).在文[3],Li首先引入算子TgCφ,并研究了该算子从H∞和Bloch空间到Zygmund空间的有界性和紧性.之后,Li和Stevic在文[10]又研究了该算子在Bloch空间上的有界性和紧性.本文主要研究TgCφ在单位球Bn上从Bloch型空间到Zygmund型空间的算子有界性和紧性.文[3]中的部分结果正好是本文在n=1,ω (r)=1-r2时的结果.
本文涉及的C表示正常数,在不同的位置可表示不同的数.A⋍B表示存在常数C使得
![](https://cimg.fx361.com/images/2023/0413/7a9f680eeaeea9adc19139f574f89f4d823aa3e3.webp)
1 TgCφ的有界性
为了研究TgCφ的有界性,我们先引入几个引理.引理1[6]设f∈Βω,则
![](https://cimg.fx361.com/images/2023/0413/9ab6330a9ebf2e8639a80d8e1632c8c57783bd15.webp)
![](https://cimg.fx361.com/images/2023/0413/e7d88ef3155441a3d8155bf8759f62fe40699d01.webp)
文章所出现的函数h都如这里所述.
引理2 给定正规函数ω,则h∈H(Bn),|h(z)|≤h(|z|)∈R,z∈Bn.且
![](https://cimg.fx361.com/images/2023/0413/3f07b0584b09f6840f7425a2a90f83afb485d8c3.webp)
进一步,对任意的r∈(0,1),ω(r)⋍ω(r2).
证明 可参见文[5].
定理1 设g∈H(Bn),φ是Bn上的全纯自映射.则下列各条款等价:
(i)TgCφ∶Βω→Ζμ有界;
(ii)TgCφ∶Βω,0→Ζμ有界;
![](https://cimg.fx361.com/images/2023/0413/0ceabdaef16fe0d9ca94ed02fa5f48ef416813b8.webp)
证明 (i)⇒(ii)是显然的.
(iii)⇒(i).对任意的f∈Βω,有:
![](https://cimg.fx361.com/images/2023/0413/e27ce51e6394a4004179f62cd2f558477b3989c2.webp)
![](https://cimg.fx361.com/images/2023/0413/b1eaee0149a90d08d268bd7ab4ef326926f3f05e.webp)
由(2)、(3)、(4)、(6)、(7)、(9)式得:
![](https://cimg.fx361.com/images/2023/0413/af980181ecf462e7f4ced22031be3f3932630e07.webp)
结合(8)式得:
![](https://cimg.fx361.com/images/2023/0413/2bf476d144a44531cc824c0f45c52ad7002e966c.webp)
(i)⇒(iii).首先,假设TgCφ∶Βω,0→Ζμ有界.令f(z)=1,则(f°φ)(z)=1,ℜ(f°φ)(z)=0,由(8)、(2)式及TgCφ的有界性得:
![](https://cimg.fx361.com/images/2023/0413/8c84decf267958c5d82ce51939d83b935d07f2ba.webp)
再令f(z)=z,由(8)、(2)、(10)式知:
![](https://cimg.fx361.com/images/2023/0413/95aa2a84cb5088e2af170c0a7101725ee01b9956.webp)
对任意w∈Bn,令
![](https://cimg.fx361.com/images/2023/0413/cb668582044e0622d18ac8735e13220d337bec30.webp)
![](https://cimg.fx361.com/images/2023/0413/eccce268178b3c70261fa215ec8a848fff8f9151.webp)
其次,设w∈Bn,当|φ(w)|≤δ,δ∈(0,1),显然有:
![](https://cimg.fx361.com/images/2023/0413/22c292698c5d2030c30ccbaa40ec6edef4b1d675.webp)
当δ<|φ(w)|<1时,令
![](https://cimg.fx361.com/images/2023/0413/24750df225a5284c82351967c9ea58e2db9aabf9.webp)
![](https://cimg.fx361.com/images/2023/0413/3f09d8762d3334dc12d7f5c9078ccca2db0102ba.webp)
由引理2知,
![](https://cimg.fx361.com/images/2023/0413/a3b264591c7e970758e3f01c2ef12f049538a74a.webp)
由TgCφ∶Βω,0→Ζμ有界.故对任意的w∈Bn,有:
![](https://cimg.fx361.com/images/2023/0413/f36a39cf6d9434087a79f0b52398a9f6b08d1676.webp)
定理证完.
定理2 设g∈H(Bn),φ是Bn上的全纯自映射,则TgCφ∶Βω,0→Ζμ,0有界的充要条件是TgCφ∶Βω,0→Ζμ有界,且
![](https://cimg.fx361.com/images/2023/0413/512e2836b3970fa7e1616e17ad41891b7d592d0a.webp)
成立.
证明 必要性 假设TgCφ∶Βω,0→Ζμ,0有界,则TgCφ∶Βω,0→Ζμ有界,且对任意的f∈Βω,0有TgCφf∈Ζμ,0.取f(z)=1,则
![](https://cimg.fx361.com/images/2023/0413/cbeca1e51d0116e9783c587691f9f58ebf0b291b.webp)
取f(z)=z,则
![](https://cimg.fx361.com/images/2023/0413/d928ff3b02d57cbab9fa7bb04aa49bcb25c720ce.webp)
所以(12)、(13)式成立.
充分性 设p是一多项式,由(13)、(14)式知,当|z|→1时,
![](https://cimg.fx361.com/images/2023/0413/dc484e2ea45294775daed166d1b62079b0d29b19.webp)
所以TgCφp∈Ζμ,0.又因为多项式集是Βω,0中的稠密集,所以对任意的f∈Βω,0,存在多项式列{pk}k∈N*,使得‖Pk-f‖Βω→0,(k→∞).于是,对任意的f∈Βω,0,当k→∞时,由TgCφ∶Βω,0→Ζμ的有界性,
![](https://cimg.fx361.com/images/2023/0413/dcff1e37403b7141f65f73aec4b72eb9033c2870.webp)
所以TgCφf∈Ζμ,0.由闭图像定理,所以TgCφ∶Βω,0→Ζμ,0有界.定理证毕.
2 TgCφ的紧性
引理3 闭集K⊂Ζμ,0是紧集当且仅当K有界且满足:
![](https://cimg.fx361.com/images/2023/0413/d75e77350ba92968fb5a2326c1cf43982da317c7.webp)
证明 类似于文[11]中引理1.
引理4 设g∈H(Bn),φ是Bn上的全纯自映射,有界算子TgCφ∶Βω(Βω,0)→Ζμ是紧算子当且仅当对Βω(Βω,0)中任意有界且在Bn内内闭一致收敛于零的序列{fk}k∈N*有‖TgCφfk‖μ→0(k→∞).
定理3 设g∈H(Bn),φ是Bn上的全纯自映射,TgCφ∶Βω→Ζμ有界.则下列各条款等价:
(i)TgCφ∶Βω→Ζμ是紧算子;
(ii)TgCφ∶Βω,0→Ζμ是紧算子;
![](https://cimg.fx361.com/images/2023/0413/cdc605c142897d1b7ee99c0c3833a59338e021de.webp)
证明 (iii)→(i).设(15)、(16)式成立,则对任意的ε>0,存在δ∈(0,1),当δ<|φ(z)|<1时,有:
![](https://cimg.fx361.com/images/2023/0413/da36fd5cac758488b83b82fed78ab417ec279463.webp)
![](https://cimg.fx361.com/images/2023/0413/58b0d74175ee2fa0a33f84d944332392e5cc00e6.webp)
由于{fk}与{▽fk}在Bn内内闭一致收敛以及ε的任意性知,当k→∞时,
![](https://cimg.fx361.com/images/2023/0413/23b3a2663540200c4b59f8ae4b873d5b95d90e3c.webp)
由引理4知TgCφ∶Βω→Ζμ是紧算子.
(ii)→(iii).事实上,(16)式的成立等价于
![](https://cimg.fx361.com/images/2023/0413/2e8357cbf319bb056051170f36f1acc86892cbdb.webp)
设{zk}k∈N*是Bn中的序列,满足|φ(zk)|→1(k→∞).令
![](https://cimg.fx361.com/images/2023/0413/55a269e5b5cd59523ded9a6787e46799b7acc271.webp)
则fk∈Βω,0,‖fk‖Βω≤C.且
![](https://cimg.fx361.com/images/2023/0413/5fda142740aaf5699c86c67d27ee807c1cddd88a.webp)
由(5)和(2)式得:
![](https://cimg.fx361.com/images/2023/0413/435951dfabf2ae5daf0333d87d01b344b4631b44.webp)
由TgCφ的紧性可得(16)式.
对上述的{zk},令
![](https://cimg.fx361.com/images/2023/0413/9da013b24d533be441eac484adbe8ba35dadbf26.webp)
![](https://cimg.fx361.com/images/2023/0413/7b0bcbbd4a4d0b926dbaf3008641413c72f8cad7.webp)
于是,由引理2得:
![](https://cimg.fx361.com/images/2023/0413/7f7927e718a84000fd999d1d15609d455e739cbd.webp)
由于TgCφ的紧性和(16)式成立,得:
![](https://cimg.fx361.com/images/2023/0413/4f61c71efadf166cd4fba48d918a36434abe03f5.webp)
定理证毕.
定理4 设g∈H(Bn),φ是Bn上的全纯自映射,则下列各条款等价:
(i)TgCφ∶Βω→Ζμ,0是紧算子;
(ii)TgCφ∶Βω,0→Ζμ,0是紧算子;
![](https://cimg.fx361.com/images/2023/0413/9646577676e7584e2ebb206447642527b5beac08.webp)
证 (i)→(ii)是显然的.
(iii)→(i).由引理3知,TgCφ∶Βω→Ζμ,0是紧算子当且仅当
![](https://cimg.fx361.com/images/2023/0413/fa1fa9c246a962a4bd0f899009943c5efa6f6ee2.webp)
设‖f‖Βω≤1,由(19)、(20)式得:
![](https://cimg.fx361.com/images/2023/0413/d408ae027210bb1d9a69c15018598f60f2ac06da.webp)
所以TgCφ∶Βω→Ζμ,0是紧的.
(ii)→(iii).假设TgCφ∶Βω,0→Ζμ,0是紧的,则TgCφ∶Βω,0→Ζμ是紧算子.由定理3(15)式知,对任意的ε>0,存在δ∈(0,1),使得当δ<|φ(z)|<1时,有:
![](https://cimg.fx361.com/images/2023/0413/8569aac2027f0d80ee0294c7df0b79c731e1774a.webp)
因为TgCφ∶Βω,0→Ζμ,0是紧算子,所以TgCφ∶Βω,0→Ζμ,0有界,由定理2知:
![](https://cimg.fx361.com/images/2023/0413/71986b1871e42ef85d5e421b35fe4b779124dba1.webp)
故对上述的ε,存在r∈(0,1),使得当r<|z|<1时,有:
![](https://cimg.fx361.com/images/2023/0413/2a196c409ac85b9f4b2f5ae2326e4070c627324d.webp)
因此,当δ<|φ(z)|<1,r<|z|<1时,
![](https://cimg.fx361.com/images/2023/0413/9c454f38adc12c578d73e9930dddc9e3effa980d.webp)
当|φ(z)|≤δ,r<|z|<1时,
![](https://cimg.fx361.com/images/2023/0413/9af3517e7556e0ce439b30c73dafb4782ae4fbea.webp)
由(21)、(22)式及ε的任意性可得(15)式.同样的方法可得到(16)式.证毕.
致谢:衷心感谢胡璋剑教授的精心指导.
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Products of ExtendedCesàroOperator and Composition Operator fromBloch-typeSpaces toZygmund-typeSpaces
OU YANG Xiao-rong
(College of Mathematica,Physics and Information Engineering,Zhejiang Normal University,Jinhua 321004,China)
Letωandμbe normal function,gbe holomorphic function on the unit ball andφbe holomorphic self-mapping ofBn.The operatorTgCφ∶Bω(Bω,0)→Zμ(Zμ,0)induced bygandφ,defined byTgCφfditions for the operatorTgCφfromBloch-ty pespaces toZy gmund-ty pespaces.
Bloch-ty pespaces;Zy gmund-ty pespaces;extended Cesàrooperator;composition operator
O175.14
A
1009-1734(2011)01-0018-07
2010-10-24;
2010-11-12
国家自然科学基金项目(10771064);浙江省自然科学基金项目(Y7080197,Y6090036,Y6100219).
欧阳小荣,浙江师范大学数理信息与工程学院2008级在读硕士,从事函数论研究.
MSC 2000:47B38