Dirichlet空间上Toeplitz算子与Berezin型变换相关的一些性质

Some properties of Toeplitz operators on Dirichlet space with respect to Berezin type transform

本文研究了单位圆盘D的Dirichlet空间上Toeplitz算子和小Hankel算子.利用Berezin型变换讨论了Toeplitz算子的不变子空间问题,具有Berezin型符号的Toeplitz算子的渐进可乘性以及Toeplitz算子的Riccati方程的可解性.应用Berezin变换得到了Toeplitz算子和小Hankel算子可逆的充分条件.此外,还利用Hankel算子和Berezin变换刻画了算子2TuvTuTvTvTu的紧性,其中函数u,v∈L2,1.

In this paper, we consider the Toeplitz operators and little Hankel operators on the Dirichlet space D of the unit disc D. By Berezin type transform, the invariant subspaces of Toeplitz operators, asymptotic multiplicative property of the Berezin type symbols of Toeplitz operators and solvability of the Riccati equations of Toeplitz operators are discussed. We also obtained a sufficient condition for invertibility of Toeplitz operators and little Hankel operators in terms of Berezin transform. Moreover, the Hankel operators and Berezin transform are used to characterize compactness of operators 2Tuv-TuTv-TvTu for functions u and v in L2,1 .