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

    2012-12-25 06:49:22ZHAODi
    湖州師范學院學報 2012年1期
    關鍵詞:浙江師范大學數(shù)理本性

    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型空間到其他空間上的廣義Cesàro算子的本性模

    趙 弟

    (浙江師范大學 數(shù)理與信息工程學院,浙江 金華321004)

    H(B)表示單位球上B的全純函數(shù)類.對p≥0,單位球上的Bloch型空間用Bp表示.對給定的g∈H(B),我們給出了廣義Cesàro算子Tg在不同Bloch型空間上本性模的等價條件.

    廣義Cesàro算子;Bloch型空間;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.

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