Toeplitz算子的特征值和特征向量  

The Eigenvalues and Eigenvectors of the Toeplitz Operators

作　　者：丁宣浩[1,2] 侯林 李永宁 Xuan Hao DING;Lin HOU;Yong Ning LI(School of Mathematics and Statistics,Chongqing Technology and Business University,Chongqing 400067,P.R.China;Chongqing Key Laboratory of Social Economy and Applied Statistics,Chongqing 400067,P.R.China)

机构地区：[1]重庆工商大学数学与统计学院,重庆400067 [2]经济社会应用统计重庆市重点实验室,重庆400067

出　　处：《数学学报（中文版）》2023年第1期181-186,共6页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金(11871122,12101092);重庆市自然科学基金(cstc2020jcyj-msxmX0318);重庆市教委基金(KJQN202100822);重庆工商大学基金(2020316,2053010)。

摘　　要：记L^(2)为单位圆周T上的Lebesgue平方可积函数全体.定义Hardy空间H^(2)是L^(2)中的解析多项式所张成的闭子空间.对于复平面中的单位开圆盘D中的任一点z,函数K_(z)(w)=1/(1-zw)是H^(2)中的再生核函数.对任意的f∈H^(2),众所周知T_(f)K_(z)=f(_(z))K_(z),即K_(z)是T_(f)的属于特征值f(z)的特征向量.反过来,若存在z∈D(或对每一个z∈D),使得K_(z)是T_(f)的特征向量,是否必有f∈H∞针对这些问题,本文给出了以再生核K_(z)为特征向量的Toeplitz算子以及有界线性算子的完全刻画,还给出了以所有的f(z)(z∈D)为特征值的Toeplitz算子的部分刻画.Let L^(2) be the set consisting of all the Lebesgue integrable functions on the unit circle T.Define the Hardy space to be the closed subspace spanned by the analytic polynomials in L^(2).For any point z in the unit disk D of the complex plane,Kz(w)=1/(1-zw)is the reproducing kernel function in H^(2).It is well known that Tf Kz=f(z)Kz,that is,Kz is the eigenvector of Tfcorresponding to the eigenvalue f(z).Conversely,if there exists some z∈D(or,for any z∈D),K_(z) is an eigenvector of Tf,whether there must be f∈H∞For the above questions,in this paper,we give a complete characterization of the Toeplitz operators as well as the bounded linear operators which take the reproducing kernels Kz as their eigenvectors.Moreover,we partially describe the Toeplitz operators with f(z)(z∈D)as their eigenvalues.

关 键 词：HARDY空间 TOEPLITZ算子 Coburn引理 特征值 

分 类 号：O177.2[理学—数学]