Dirichlet空间上Toeplitz算子的一些性质

该文研究了Dirichlet空间D^p~(1< p<∞)上Toeplitz算子的紧性与Fredholm性质, 计算了D^p上Toeplitz算子的Fredholm指标. 还考查了D^p上Hankel算子紧性.     

球上Zygmund 型空间上的加权Cesàro 算子

该文在Cⁿ中单位球上讨论了Zygmund 型空间(小Zygmund 型空间)之间的加权Cesàro 算子T_g 的有界性和紧性特征, 得到了以下的结果: (1) T_g 是Z^p 到Z^q的有界算子或紧算子的充要条件; (2) T_g 是 Z^p₀ 到Z^q₀ 的有界算子或紧算子的充要条件.     

用于边缘检测的Laplace——样条(▽2S)算子

根据数据平滑理论,提出了一种图象边缘检测的方法:把二维四阶B样条函数与Laplacian算子相结合组合成▽²S算子,对图象卷积之后,根据过零点(ZeroCrossing)确定图象边缘。文中证明了它在图象平滑和边缘检测效果上,优于Marr和Hildreth提出的▽²G算子,且计算量较小。而▽²G算子可用高阶▽²S算子近似。文中给出了一些实验结果。     

重调和Hardy空间上的Toeplitz 算子

与调和Bergman 空间相对应, 本文研究重调和Hardy 空间h²(D²) 上的Toeplitz 算子. 本文发现, h²(D²) 上的Toeplitz 算子与经典的Hardy 空间、Bergman 空间及调和Bergman 空间上的Toeplitz算子的性质都有很大的差异. 例如解析Toeplitz 算子可以不是半可换及可交换的. 即使半可换, 其中任何一个符号可以不为常数; 即使可交换, 两个符号的非平凡线性组合也不一定是常数. 本文得到了h²(D²) 上两个解析Toeplitz 算子半可换和可换的充分必要条件.     

广义ω-Calderón-Zygmund算子的有界性

本文证明了广义ω-Calderón-Zygmund算子是HA_ω^p到HA^p的有界算子.     

p-adic 域上的拟微分算子

本文研究p-adic 域Q_p 上的一类拟微分算子{T^α : α∈R}, 其中算子T^α 在Q_p 的检验函数空 间S(Q_p) 以及相应的分布空间S′(Q_p) 中作为一个运算是封闭的. 本文构造了算子T^α 的卷积核, 指 出算子与其卷积核在解决p-adic 域上的相关问题中所起的重要作用. 最后研究算子T^α 的谱性质, 构 造了算子T^α 的全体固有值与固有函数, 并证明全体固有函数所成的集合组成L²(Q_p) 空间中的一组 完整正交基.     

关于复平面上H∞空间,βp空间以及β0p空间的几个问题

本文讨论了单位圆中Hardy空间H^∞到p-Bloch空间β^p的复合算子T_1,φ加权复合算子T_ψ,φ的有界性,也讨论了H^∞到小p-Bloch空间β₀^p的复合算子T_1,φ的有界性问题;另外还讨论了小p-Bloch空间到H^∞空间的点乘子及小p-Bloch空间上复合算子的紧性等.     

Cn中球型带权Bergman空间上Toeplitz与Hankel算子之紧性

本文给出Cⁿ中单位球上的带权的Bergman空间上具L^∞-符号的Toeplitz算子或Hankel算子为紧的充要条件。     

H2(Bn)上的Toeplitz算子与复合算子

给出H²(B_n)上复合算子具有闭值域的特征,讨论了Toeplitz算子与复合算子的Fredholm性质。     

关于广义Calderon－Zygmund算子

本文讨论了θ（ｔ）型和（ｌｏｇ，θ）型Ｃａｌｄｅｒóｎ－Ｚｙｇｍｕｎｄ算子在加权Ｈａｒｄｙ型空间ＨＡ_w^p上的有界性，θ（ｔ）型Ｃａｌｄｅｒóｎ－Ｚｙｇｍｕｎｄ算子在Ｈａｒｄｙ型加权块空间上的有界性，以及广义的ｗ－Ｃａｌｄｅｒóｎ－Ｚｙｇｍｕｎｄ算子是Ｈ^Ap_w到ＨＡ^p上的有界算子．     

Fock空间上径向函数诱导的Toeplitz算子

本文讨论了Fock空间上以径向函数和拟齐次函数为符号的Toeplitz算子的代数性质,给出了两个以径向函数为符号的Toeplitz算子的积仍为Toeplitz算子的充分必要条件,并且研究了以拟齐次函数为符号的Toeplitz算子的交换性.     

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.     

Toeplitz Operators on the Fock Space

We study Toeplitz operators on the Fock space with positive measures as symbols. Main results include characterizations of Fock–Carleson measures, bounded Toeplitz operators, compact Toeplitz operators, and Toeplitz operators in the Schatten p-classes.     

Algebraic Properties of Toeplitz Operators on the Harmonic Dirichlet Space

We study some algebraic properties of Toeplitz operators on the harmonic Dirichlet space of the unit disk. We first give a characterization for boundedness of Toeplitz operators. Next we characterize commuting Toeplitz operators. Also, we study the product problem of when product of two Toeplitz operators is another Toeplitz operator. The corresponding problems for compactness are also studied.     

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.     

Dual Toeplitz operators on the sphere

Dual Toeplitz operators on the Hardy space of the unit circle are anti-unitarily equivalent to Toeplitz operators. In higher dimensions, for instance on the unit sphere, dual Toeplitz operators might behave quite differently and, therefore, seem to be a worth studying new class of Toeplitz-type operators. The purpose of this paper is to introduce and start a systematic investigation of dual Toeplitz operators on the orthogonal complement of the Hardy space of the unit sphere in Cn . In particular, we establish a corresponding spectral inclusion theorem and a Brown-Halmos type theorem. On the other hand, we characterize commuting dual Toeplitz operators as well as normal and quasinormal ones.     

Asymptotic Toeplitz and Hankel operators on the Bergman space

In this paper the concept of asymptotic Toeplitz and asymptotic Hankel operators on the Bergman space are introduced and properties of these classes of operators are studied. The importance of this notion is that it associates with a class of operators a Toeplitz operator and with a class of operators a Hankel operator where the original operators are not even Toeplitz or Hankel. Thus it is possible to assign a symbol to an operator that is not Toeplitz or Hankel and hence a symbol calculus is obtained. Further a relation between Toeplitz operators and little Hankel operators on the Bergman space is established in some asymptotic sense.     

单位球上的对偶Toeplitz算子

本文研究了单位球Bergman空间的直交补上的对偶Toeplitz算子的代数性质,首先我们给出了对偶Toeplitz算子的有界性和紧性的完全刻画,然后给出对偶Toeplitz算子的谱性质,最后证明了不存在以有界全纯或者反全纯函数为符号的拟正规对偶Toeplitz算子.     

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.     

Commuting Dual Toeplitz Operators on the Polydisk

On the polydisk, the commutativity of dual Toeplitz operators is studied. We obtain characterizations of commuting dual Toeplitz operators, essentially commuting dual Toeplitz operators and essentially semi-commuting dual Toeplitz operators.