Essential Norm of Extended Cesàro Operators from One Bloch-type Space to Another*

ZHAO Di

（Department of Mathematics，Zhejiang Normal University，Jinhua 321004，China）

Essential Norm of Extended Cesàro Operators from One Bloch-type Space to Another*

ZHAO Di

（Department of Mathematics，Zhejiang Normal University，Jinhua 321004，China）

H（B）denotes the family of all holomorphic function on the unit ball B.For p≥0，Bpdenotes the Blochtype space on the unit ball B.In this paper，we give the equivalent conditions of the essential norm of the extended Cesàro operator Tgon Bloch-type space，where g is a given holomorphic function on the unit ball B.

extended Cesàro operator；Bloch-type space；essential norm

MSC 2000：47B38 32A36

0 Introduction

In the unit ball，the boundedness and compactness of the extended Cesàro operators have been considered by many authors.Hu［3］discussed the boundedness and compactness of Tgon the Bergman space.Hu［4］got the characterization on g for which the induced extended Cesàro operator is bounded or compact onand.Zhang［5］obtained the sufficient and necessary conditions on g such that Tgis bounded or compact betweenandfor 0≤p，q＜∞.The essential norm of composition operator on Bergman spaces，Bloch-type spaces and Hardy spaces have been studied in ［6～9］.Recently，Hu［10］discussed the essential norm of the extended Cesàro operator from one Bergman space to anoher.The purpose of this work is to obtain the essential norm of Tgbetweenandfor 0≤p，q＜∞.

In what follows，C will stand for positive constants whose value may change form line to line but does not depend on the functions in H（B）.The expression A⋍B means C－1≤B≤CA.

1 Some preliminary results

［1］YANG W S，OUYANG C H.Exact location ofα-Bloch spaces in Lapand Hpof a complex unit ball［J］.Rochy Mountain J Math，2000，30：1151～1169.

［2］ZHOU W X，YANG W S.Integral criteria of Bloch type functions on radial derivative［J］.Acta Math Sci，1998，18：451～458.

［3］HU Z J.Extended Cesàro operators on Bergman spaces［J］.J Math Anal Appl，2004，296：435～454.

［4］HU Z J.Extended Cesàro operators on the Bloch space in the unit ball of Cn［J］.Acta Math Sci ser B，2003，23（4）：561～566.

［5］ZHANG X J.Weighted Cesàro operators on Dirichlet type spaces and Bloch spaces of Cn（Chinese）［J］.Chin Ann Math ser A，2005，26（1）：139～150.

［6］Cuckovicz，ZHAO R H.Weighted composition operators on the Bergman space［J］.J London Math Soc，2004，70（2）：499～511.

［7］LUO L.The essential norm of a composition operator on the Bergman spaces of the unit ball［J］.Complex Var Theory Appl，2004，49（12）：845～850.

［8］HU S H.The essential norm of a composition operator on the Bloch space over the unit ball in Cn［J］.Chinese Acta Math Sinica，2007，50（5）：1151～1160.

［9］LUO L，SHI J H.The essential norm of a composition operator on Hardy spaces of the unit ball［J］.Chinese Ann Math Ser A，2007，28（6）：805～810.

［10］HU Z J.Essential norm of extended Cesàro operators form one Bergman space to another［J］.Bull Austral Math Soc，2012，85：307～314.

［11］ZHU K H.Spaces of holomorphic functions in the unit ball ［M］.New York：Graduate Texts in Mathematics 226，Springer-Verlag，2005：43～44.

［12］RUDIN W.Function theory in the unit ball of Cn［M］.New York：Springer-Verlag，1980：17～18.

［13］SHIELDS A L，WILLIAMS D L.Bound projections，duality，and multiplies in spaces of analytic functions［J］.Trans Amer Math Soc，162（1971）：287～302.

MSC 2000：47B38

Bloch型空间到其他空间上的广义Cesàro算子的本性模

赵 弟

（浙江师范大学 数理与信息工程学院，浙江 金华321004）

H（B）表示单位球上B的全纯函数类.对p≥0，单位球上的Bloch型空间用Bp表示.对给定的g∈H（B），我们给出了广义Cesàro算子Tg在不同Bloch型空间上本性模的等价条件.

广义Cesàro算子；Bloch型空间；本性模

O174.56

O174.56 Document code：A Article ID：1009-1734（2012）01-0016-07

date：2011-12-06

s：Supported by the National Natural Science Foundation of China（10771064），Natural Science Foundation of Zhejiang Province（Y7080197，Y6090036，Y6100219），and Foundation of Creative Group in Colleges and U-niversities of Zhejiang Province（T200924）.

Biography：ZHAO Di，graduate student 2009 of Zhejiang Normal University，engages in complex variables and operator theory.