Dirichlet空间上的Toeplitz算子

本文研究了Dirichlet空间上的Toeplitz算子,部分的回答了文[1]中的问题,给出了关于Dirichlet空间上Toeplitz算子的一个稠密性定理。     

圆环上的Dirichlet空间及其算子

本文主要讨论了圆环上Dirichlet空间中Toeplitz算子的Fredholm性质,计算了符号在C~1和H_1~∞+C~1中的Toeplitz算子的本性谱.     

Toeplitz Algebras on Dirichlet Spaces

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　ＬｅｔＢｎｂｅｔｈｅｕｎｉｔｂａｌｌｉｎＣｎ,ａｎｄｄｖｔｈｅｎｏｒｍａｌｉｚｅｄＬｅｂｅｓｇｕｅｍｅａｓｕｒｅｏｎＢｎ.ＳｏｂｏｌｅｖｓｐａｃｅＬ2 ,1ｉｓｔｈｅｓｐａｃｅｏｆｆｕｎｃｔｉｏｎｓｕ :Ｂｎ→Ｃ ,ｆｏｒｗｈｉｃｈｔｈｅｎｏｒｍ‖ｕ‖1/ 2 =∑ｎｉ=1 ∫Ｂｎ ｕ ｚｉ(ｚ) 2 + ｕ ｚｉ(ｚ) 2 ｄν1/ 2 <∞ .Ｌ2 ,1/ 2 ｉｓａＨｉｌｂｅｒｔｓｐａｃｅｗｉｔｈｔｈｅｉｎｎｅｒｐｒｏｄｕｃｔ〈ｕ ,ｖ〉1/ 2 =∑ｎｉ=1〈 ｕ ｚｉ, ｕ ｚｉ〉Ｌ2 (ｄν) +∑ｎｉ=1〈 ｕ ｚｉ, ｕ …     

Dirichlet空间上Toeplitz算子的紧性

本文给出了 Ｄｉｒｉｃｈｌｅｔ空间上 Ｔｏｅｌｐｉｔｚ算子为紧算子的充要条件,并证明具有 Ｃ１－符号的 Ｔｏｅｐｌｉｔｚ算子为紧算子当且仅当它为零算子,当且仅当符号的边值为零．     

Properties of Toeplitz Operators on the Dirichlet Space Over the Ball

On the Dirichlet space of the unit ball, we study some algebraic properties of Toeplitz operators. We give a relation between Toeplitz operators on the Dirichlet space and their analogues defined on the Hardy space. Based on this, we characterize when finite sums of products of Toeplitz operators are of finite rank. Also, we give a necessary and sufficient condition for the commutator and semi-commutator of two Toeplitz operators being zero.     

Dirichlet空间上的Bergman型Toeplitz算子

本文给出了Dirichlet空间上以有界调和函数为符号的Bergman型Toeplitz算子是紧算子的充要条件.同时刻画了此类Bergman型Toeplitz算子在Dirichlet空间上的交换性.     

Algebraic properties of Toeplitz operators on the Dirichlet space

We study some algebraic properties of Toeplitz operators on the Dirichlet space. We first characterize (semi-)commuting Toeplitz operators with harmonic symbols. Next we study the product problem of when product of two Toeplitz operators is another Toeplitz operator. As an application, we show that the zero product of two Toeplitz operators with harmonic symbol has only a trivial solution. Also, the corresponding compact product problem is studied.     

Product of Toeplitz operators on the harmonic Dirichlet space

In this paper, we study Toeplitz operators with harmonic symbols on the harmonic Dirichlet space, and show that the product of two Toeplitz operators is another Toeplitz operator only if one factor is constant.     

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

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2T_uv-T_uT_v-T_vT_u 的紧性, 其中函数u,v ∈ L^2,1.     

Bergman空间上Toeplitz算子的局部化

In this paper we characterize the essential spectra of Toeplitz operators on Bergman spaces by using Douglas‘ localization Theorem and obtain the local decomposition of the essential spectra of Toeplitz operators.     

有界对称区域上Dirichlet空间中的紧Toeplitz算子

介绍了有界对称区域Ω上Dirichlet空间中的Toeplitz算子的紧性:如果S是有限个Toeplitz算子乘积的有限和,S是紧算子当且仅当S的Berezin变换S(z)趋于0.     

Dirichlet空间上Toeplitz算子的亚正规性

讨论单位圆盘中Dirichlet空间上Toeplitz算子的性质,给出了Dirichiet空间上以一类连续函数为符号的Toeplitz算子满足亚正规性的充分必要条件．     

Toeplitz Operators on Arveson and Dirichlet Spaces

We define Toeplitz operators on all Dirichlet spaces on the unit ball of and develop their basic properties. We characterize bounded, compact, and Schatten-class Toeplitz operators with positive symbols in terms of Carleson measures and Berezin transforms. Our results naturally extend those known for weighted Bergman spaces, a special case applies to the Arveson space, and we recover the classical Hardy-space Toeplitz operators in a limiting case; thus we unify the theory of Toeplitz operators on all these spaces. We apply our operators to a characterization of bounded, compact, and Schatten-class weighted composition operators on weighted Bergman spaces of the ball. We lastly investigate some connections between Toeplitz and shift operators. The research of the second author is partially supported by a Fulbright grant.     

Dirichet空间上Toeplitz算子的若干问题

讨论了Dirichlet空间上Toeplitz算子的紧性,特别地得到了Schatlen类Toeplitz算子的特征.此外,还证明了关于Toeplitz算子的一个非稠密性定理,并证明一个非零的函数可以诱导一个零算子,这与Hardy空间及Bergman空间情形是一重大差别.     

Schatten-Herz type positive Toeplitz operators on pluriharmonic Bergman spaces

Recently, Schatten-Herz type Toeplitz operators have been studied on the Bergman spaces and the harmonic Bergman spaces. Motivated these results, we study characterizations of positive Toeplitz operators of Schatten-Herz type in terms of averaging functions and Berezin transforms of symbol functions on the ball of pluriharmonic Bergman spaces.     

Commuting dual Toeplitz operators on the orthogonal complement of the Dirichlet space

In this paper we characterize commuting dual Toeplitz operators with harmonic symbols on the orthogonal complement of the Dirichlet space in the Sobolev space. We also obtain the sufficient and necessary conditions for the product of two dual Toeplitz operators with harmonic symbols to be a finite rank perturbation of a dual Toeplitz operator.     

Composition operators on weighted Dirichlet spaces

We characterize bounded and compact composition operators on weighted Dirichlet spaces. The method involves integral averages of the determining function for the operator, and the connection between composition operators on Dirichlet spaces and Toeplitz operators on Bergman spaces. We also present several examples and counter-examples that point out the borderlines of the result and its connections to other themes.

     

Operators on the orthogonal complement of the Dirichlet space

In this paper we prove that a dual Hankel operator is zero if and only if its symbol is orthogonal to the Dirichlet space in the Sobolev space, and characterize the symbols for (semi-)commuting dual Toeplitz operators on the orthogonal complement of the Dirichlet space in Sobolev space.     

Compact Toeplitz operators on the weighted Dirichlet space

In this paper, we completely characterize the compactness of Toeplitz operators with continuous symbol on the weighted Dirichlet space.