加权Dirichlet空间上紧Toeplitz算子

Compact Toeplitz Operators on Weighted Dirichlet Space

对α>-1,若算子S是加权Dirichlet空间Dα上有限个Toeplitz算子乘积的有限和,利用不同于加权Dirichlet空间再生核的一种新奇异积分核,得到了S为紧算子的充要条件是当z趋于单位圆盘边界时,S的类Berezin变换趋于0.又利用与Bermgan空间不同的酉算子Uz,定义了算子乘积Sz=UzSUz,得到S为紧算子的充要条件是当z趋于单位圆盘边界时,Szw在D内弱收敛到0.

Let α-1 and S be a finite sum of finite products of Toeplitz operators in a weighted Dirichlet space Dα.By using a new singular integral kernel which is different from the recovery kernel in weighted Dirichlet space,it is proved that S is compact if and only if the Berezin transformation tends to 0 as z goes to the boundary of the unit disk.Furthermore,an operator product Sz=UzSUz is defined,where Uz is the unitary operator which is different from that in Bergman space.It is proved that S is compact if and only if Szw is weakly converged to 0 as z goes to the boundary of the unit disk.