Bloch型空间上的广义Hilbert算子  

Generalized Hilbert Operator Acting on Bloch Type Spaces

作　　者：叶善力[1] 周智慧 Shan Li YE;Zhi Hui ZHOU(Zhejiang University of Science and Technology,School of Science,Hangzhou 310023,P.R.China)

机构地区：[1]浙江科技学院理学院,杭州310023

出　　处：《数学学报（中文版）》2023年第3期557-568,共12页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(11671357,11801508);浙江省自然科学基金资助项目(LY23A010003)。

摘　　要：设μ是区间[0,1)上的正Borel测度.对α>0,定义一广义的Hilbert矩阵H_(μ,α)=(μ_(n,k,α))_(n,k≥0),其中μ_(n,k,α)=∫_([0,1))Г(n+α)/n!Г(α)t^(n+k)dμ(t).通过该矩阵作用于单位圆盘D上的解析函数f(z)=∑_(k=0)^(∞)(∑_(k=0)^(∞)a_(k)z^(k))的泰勒系数,可定义一广义的Hilbert算子H_(μ,α),使得H_(μ,α)(f)(z)=∑_(n=0)^(∞)(∑_(n=0)^(∞)μ_(n,k,a)a_(k))z^(n).本文给出广义的Hilbert算子H_(μ,α)(α≥2)是Bloch型空间B_(β)(0<β<∞)到B_(α-1)空间上是有界(或紧)算子的充要条件,同时也给出H_(μ,α)(α>0)是Bloch型空间B_(β)到一般的Bloch型空间上是有界算子的一个必要条件.Let μ be a positive Borel measure on the interval[0,1).For α>0,the generalized Hilbert matrix H_(μ,α)=(μ_(n,k,α))_(n,k≥0) with μ_(n,k,α)=∫_([0,1))Г(n+α)/n!Г(α)t^(n+k)dμ(t) formally induces the generalized Hilbert operator H_(μ,α)(f)(z)=∑_(n=0)^(∞)(∑_(n=0)^(∞)μ_(n,k,a)a_(k))z^(n) on the space of l analytic functions f(z)=∑_(k=0)^(∞)(∑_(k=0)^(∞)a_(k)z^(k)) in the unit disc D.In this paper,we characterize the measures μ for which H_(μ,α)(α≥2)is a bounded(resp.,compact)operator from the Bloch type space Bp(0<β<oo)into B_(α-1).We also give a necessary condition for which H_(μ,α)(α≥2)is a bounded operator by acting on Bloch type spaces for general cases.

关 键 词：广义Hilbert算子 BLOCH型空间 CARLESON测度 

分 类 号：O174.5[理学—数学]