作用在加权Bergman空间上的无穷维Hilbert张量  

Infinite-dimensional Hilbert Tensors Acting on Weighted Bergman Spaces

作　　者：王翠[1] WANG CUI(Department of Mathematics and Statistics,Guangdong University of Technology,Guangzhou 510520,China)

机构地区：[1]广东工业大学数学与统计学院,广州510520

出　　处：《应用数学学报》2023年第5期744-750,共7页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金(11801094)资助项目。

摘　　要：本文首先介绍了一些基本的定义和事实,它们将用于证明我们的主要结果.其次,我们给出了Hilbert张量算子H的定义,并借助Song和Qi文章中的证明技巧,给出了一些引理,这些引理表明Hilbert张量算子H是良性定义的.此外,本文引入了Song和Qi给出的Hilbert张量算子的积分形式.随后,本文刻画了m阶无穷维Hilbert张量(超矩阵,即Hilbert张量算子),从加权Bergman空间Aα(p(m-1))(α>-1,α+2 <p <+∞)到Aβq(β>-1,0 <q <β+1)的有界性及(m-1)阶齐次性;TH,FH是由Hilbert张量算子H诱导出的正齐次算子,借助Hilbert张量算子H在加权Bergman空间上的有界性及齐次性,文章证明了TH从加权Bergman空间Aα(p(m-1))(α>1,α+2 <p <+∞)到Aβq(β>-1,0 <q<β+1)的有界性及正齐次性,FH从加权Bergman空间Aαp(α>-1,α+2 <p <+∞)到Aβq(β>-1,0<q<(β+1)(m-1))的有界性及正齐次性.In this paper,firstly,some basic definitions and facts are introduced,which will be used in the proof of our main results.Secondly,we give the definition of a Hilbert tensor operator H,and by the proof technique of Song and Qi,some useful lemmas are given which indicate such an operator H is well-defined.Also,the integral form of the Hilbert tensor operator is given by Song and Qi.Then,we characterize the m-order infinite dimensional Hilbert tensors(hypermatrix,that is,Hilbert tensor operators)are bounded and(m-1)order homogeneous from the weighted Bergman spaces A(m-1)to Ag,where a,β>-1,α+2<p<+0 and 0<α<β+1;T and F are both bounded and positively homogeneous operators induced by Hilbert tensor operators H.Based on the boundedness and homogeneity of Hilbert tensor operators H on weighted Bergman spaces,the paper shows that the operators T are bound pstively homogenous from weighted Bergmanspaces Asd(m-1)to Ag,whereα,β>-1,α+2<p<+oo,0<q<β+1.and the operators Fαare bounded,positively homogeneous from weighted Bergman spaces A to Ag,whereα,β>-1,α+2<p<+8,0<q<(β+1)(m-1).

关 键 词：Hilbert张量 加权BERGMAN空间 齐次性 有界性 

分 类 号：O174.5[理学—数学]