Brian Simanek
We consider the Toeplitz operator with symbol z^n+C|z|^s acting on certain weighted Bergman spaces and determine for what values of the constant C this operator is hyponormal. The condition is presented in terms of the norm of an explicit block Jacobi matrix.
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We consider operators acting on a Hilbert space that can be written as the sum of a shift and a diagonal operator and determine when the operator is hyponormal. The condition is presented in terms of the norm of an explicit block Jacobi matrix. We apply this result to the Toeplitz operator with specific algebraic symbols acting on certain weighted Bergman spaces of the unit disk and determine when such operators are hyponormal.
Hyponormal Toeplitz operators with operator valued normal symbols
本文刻画了具有算子值正规符号的次正规Toeplitz算子，并探讨了该类算子何时为解析或正规。主要贡献在于给出了此类算子的结构特征。研究问题涉及算子理论中的次正规性，方法基于符号的算子值性质分析。结果对理解Toeplitz算子的分类有重要意义。
Hyponormality of the sum of two Toeplitz operators
研究了形如wT_φ+T_ψ的算子的次正则性和可逆性，以及当Toeplitz算子符号为线性泛函时的次正则性问题。
Hyponormality and quasinormality of unbounded Toeplitz operators with non harmonic symbol on the Fock Sobolev space
本文研究了非谐振算符在福克-索波列夫空间上的本征值和准本征值本质标准，揭示了非谐振算符的准本征值与本征值之间的关系。
Hyponormal Toeplitz Operators on the Bergman Space of the Disk
本文研究单位圆盘 Bergman 空间上有界符号 Toeplitz 算子的亚正规性，针对特定符号确定了参数 $|a(t)|$ 在大 $t$ 值下的上界，并更正了前人关于算子交换子范数的笔误。
In this paper, we mainly study the hyponormality of dual Toeplitz operators on the orthogonal complement of the harmonic Bergman space. First we show that the dual Toeplitz operator with bounded symbol is hyponormal if and only if it is normal. Then we obtain a necessary and sufficient condition for the dual Toeplitz operator $S_φ$ with the symbol $φ(z) = az^{n_1}\overline{z}^{m_1} + bz^{n_2} \overline{z}^{m_2}$, $(n_1,n_2,m_1,m_2\in \mathbb {N}$ and $a,b \in \mathbb{C})$ to be hyponormal. Finally, we show that the rank of the commutator of two dual Toeplitz operators must be an even number if…
In this paper we are concerned with hyponormality and subnormality of block Toeplitz operators acting on the vector-valued Hardy space $H^2_{\mathbb{C}^n}$ of the unit circle. Firstly, we establish a tractable and explicit criterion on the hyponormality of block Toeplitz operators having bounded type symbols via the triangularization theorem for compressions of the shift operator. Secondly, we consider the gap between hyponormality and subnormality for block Toeplitz operators. This is closely related to Halmos's Problem 5: Is every subnormal Toeplitz operator either normal or analytic? We sho…
In this paper, we study hyponormal weighed composition operators on the Hardy and weighted Bergman spaces. For functions $ψ\in A(\mathbb{D})$ which are not the zero function, we characterize all hyponormal compact weighted composition operators $C_{ψ,φ}$ on $H^{2}$ and $A^{2}_α$. Next, we show that for $φ\in \mbox{LFT}(\mathbb{D})$, if $C_φ$ is hyponormal on $H^{2}$ or $A^{2}_α$, then $φ(z)=λz$, where $|λ| \leq 1$ or $φ$ is a hyperbolic non-automorphism with $φ(0)=0$ and such that $φ$ has another fixed point in $\partial \mathbb{D}$. After that, we find the essential spectral radius of $C_φ$ o…
Which hyponormal block Toeplitz operators are either normal or analytic?
本文通过分析矩阵值内函数的最大公因子与弱交换性，将 Curto-Hwang-Lee 关于有理符号块 Toeplitz 算子的结果推广至有界型符号，证明满足标量内函数条件时，平方也次正规的次正规 T_Ψ 必为正规或解析算子。
Hyponormal block Toeplitz operators with non-harmonic symbols on the weighted Bergman space
研究向量值加权 Bergman 空间中符号为 $Az^p\bar{z}^q+Bz^s\bar{z}^t$ 的块 Toeplitz 算子，给出其次正规性的两个条件。
We prove some characterizations of Schatten class Toeplitz operators on Bergman spaces of tube domains over symmetric cones for small exponents.
Hyponormal block Toeplitz operators with finite rank self-commutators
本文研究了亚正则块 Toeplitz 运算符的自交换子为有限秩的条件，证明了当符号Φ满足一定条件时，自交换子为有限秩等价于存在有限 Blaschke-Potapov 积。
In this paper, we study Toeplitz operators on the weighted harmonic Bergman spaces with nonnegative symbols, the weights we choose here are Muckenhoupt A_2 weights. Results obtained include characterizations of bounded Toeplitz operators, compact Toeplitz operators, invertible Toeplitz operators and Toeplitz operators in the Schatten classes.
A New Necessary Condition for the Hyponormality of Toeplitz Operators on the Bergman Space
本文研究Bergman空间上Toeplitz算子的hyponormal性条件，推导出一个关于符号参数的严格不等式，改进了现有结果并为多变量Toeplitz算子的联合hyponormal性条件提供基础。
One of the major goals in the theory of Toeplitz operators on the Bergman space over the unit disk $\mathbb{D}$ in the complex place $\mathbb{C}$ is to completely describe the commutant of a given Toeplitz operator, that is, the set of all Toeplitz operators that commute with it. Here we shall study the commutants of a certain class of quasihomogeneous Toeplitz operators.
In this paper, we discuss the commutativity of sums of two quasihomogeneous Toeplitz operators on the Bergman space of the unit disc. Our main result goes in the direction of the conjecture in "Bicommutants of Toeplitz operators" (by I. Louhichi and N. V. Rao. Arch. Math. 91, 2008, 256-264)
The recent theorem by D. Luecking that finite rank Toeplitz-Bergman operators must be generated by a measure consisting of finitely many point masses is carried over to the many-dimensional case.
We give estimates for the essential norms of a positive Toeplitz operator on the Bergman space of a minimal bounded homogeneous domain in terms of the Berezin transform or the averaging function of the symbol. Using these estimates, we also give necessary and sufficient conditions for the positive Toeplitz operators to be compact.
In this paper, we study quasinormal and hyponormal composition operators \W with linear fractional compositional symbol $\ph$ on the Hardy and weighted Bergman spaces. We characterize the quasinormal composition operators induced on $H^{2}$ and $A_α^{2}$ by these maps and many such weighted composition operators, showing that they are necessarily normal in all known cases. We eliminate several possibilities for hyponormal weighted composition operators but also give new examples of hyponormal weighted composition operators on $H^2$ which are not quasinormal.
We consider Toeplitz operators in Bergman and Fock type spaces of polyanalytic $L^2\textup{-}$functions on the disk or on the half-plane with respect to the Lebesgue measure (resp., on $\mathbb{C}$ with the plane Gaussian measure). The structure involving creation and annihilation operators, similar to the classical one present for the Landau Hamiltonian, enables us to reduce Toeplitz operators in true polyanalytic spaces to the ones in the usual Bergman type spaces, however with distributional symbols. This reduction leads to describing a number of properties of the operators in the title, wh…
On the density of Toeplitz operators in the Toeplitz algebra over the Bergman space of the unit ball
该研究采用量子调和分析与表示论方法，给出单位球Bergman空间上Toeplitz代数中Toeplitz算子按范数稠密的夏氏定理的新证明。
In this note we show that if two Toeplitz operators on a Bergman space commute and the symbol of one of them is analytic and nonconstant, then the other one is also analytic.
For three standard models of commutative algebras generated by Toeplitz operators in the weighted analytic Bergman space on the unit disk, we find their representations as the algebras of bounded functions of certain unbounded self-adjoint operators. We discuss main properties of these representation and, especially, describe relations between properties of the spectral function of Toeplitz operators in the spectral representation and properties of the symbols.
Schatten-Herz class Toeplitz operators on weighted Bergman spaces induced by doubling weights are investigated in this paper.
We discuss resent developments in the problem of description of finite rank Toeplitz operators in different Bergman spaces and give some applications in analysis and mathematical physics
We study the boundedness of Toeplitz operators with locally integrable symbols on Bergman spaces $A^p(Ω),$ $1<p<\infty,$ where $Ω\subset \mathbb{C}$ is a bounded simply connected domain with polygonal boundary. We give sufficient conditions for the boundedness of generalized Toeplitz operators in terms of "averages" of symbol over certain Cartesian squares. We use the Whitney decomposition of $Ω$ in the proof. We also give examples of bounded Toeplitz operators on $A^p(Ω)$ in the case where polygon $Ω$ has such a large corner that the Bergman projection is unbounded.
We study the Carleson measures and the Toeplitz operators on the class of so-called small weighted Bergman spaces, introduced recently by Seip. A characterization of Carleson measures is obtained which extends Seip's results from the unit disc of $\mathbb C$ to the unit ball of $\mathbb C^n$. We use this characterization to give necessary and sufficient conditions for the boundedness and compactness of Toeplitz operators. Finally, we study the Schatten $p$ classes membership of Toeplitz operators for $1<p<\infty$.
In this paper we study the product of Toeplitz operators on the harmonic Bergman space of the unit disk of the complex plane C. Mainly, we discuss when the product of two quasihomogeneous Toeplitz operators is also a Toeplitz operator, and when such operators commute.
Let $D$ be an irreducible bounded symmetric domain with biholomorphism group $G$ with maximal compact subgroup $K$. For the Toeplitz operators with $K$-invariant symbols we provide explicit simultaneous diagonalization formulas on every weighted Bergman space. The expressions are given in the general case, but are also worked out explicitly for every irreducible bounded symmetric domain including the exceptional ones.
In this article we characterize the boundedness and compactness of a Toeplitz-type operator on weighted Bergman spaces satisfying the so-called Bekolle-Bonami condition in terms of the Berezin transform.
Numerical range of Toeplitz and Composition operators on weighted Bergman spaces
本文完整刻画了加权Bergman空间上调和符号Toeplitz算子及加权复合算子的数值域，分类了复合算子数值域对应的集合，并研究了零的包含性及所含圆和椭圆的半径。
Commuting analytic and co-analytic Toeplitz operators on the harmonic Bergman space
本文通过构造二次型，证明了调和Bergman空间上解析与共解析Toeplitz算子交换时至少一个符号为常数，解决了1999年的开放问题。
We study algebraic properties of Toeplitz operators on Bergman spaces of polyanalytic functions on the unit disk. We obtain results on finite-rank commutators and semi-commutators of Toeplitz operators with harmonic symbols. We also raise and discuss some open questions.
In this paper, we focus on the weighted Bergman spaces $A_φ^{p}$ in $\mathbb{D}$ with $φ\in\mathcal{W}_{0}$. We first give characterizations of those finite positive Borel measures $μ$ in $\mathbb{D}$ such that the embedding $A_φ^{p}\subset L_μ^{q}$ is bounded or compact for $0<p,q<\infty$. Then we describe bounded or compact Toeplitz operators $T_μ$ from one Bergman space $A_φ^{p}$ to another $A_φ^{q}$ for all possible $0<p,q<\infty$. Finally, we characterize Schatten class Toeplitz operators on $A_φ^{2}$.
We study Toeplitz operators with uniformly continuous symbols on generalized harmonic Bergman spaces of the unit ball in $\mathbb{R}^n$. We describe their essential spectra and establish a short exact sequence associated with the $C^{*}$-algebra generated by these operators.
Compared with harmonic Bergman spaces, this paper introduces a new function space which is called the pluriharmonic Hardy space $h^{2}(\mathbb{T}^{2})$. We character (semi-) commuting Toeplitz operators on $h^{2}(\mathbb{T}^{2})$ with bounded pluriharmonic symbols. Interestingly, these results are quite different from the corresponding properties of Toeplitz operators on Hardy spaces, Bergman spaces and harmonic Bergman spaces. Our method for Toeplitz operators on $h^{2}(\mathbb{T}^{2})$ gives new insight into the study of commuting Toeplitz operators on harmonic Bergman spaces.
If $μ$ is a finite measure on the unit disc and $k\ge 0$ is an integer, we study a generalization derived from Englis's work, $T_μ^{(k)}$, of the traditional Toeplitz operators on the Bergman space $A^2$, which are the case $k=0$. Among other things, we prove that when $μ\ge 0$, these operators are bounded if and only if $μ$ is a Carleson measure, and we obtain some estimates for their norms.
We characterize operator-theoretic properties (boundedness, compactness, and Schatten class membership) of Toeplitz operators with positive measure symbols on Bergman spaces of holomorphic hermitian line bundles over Kähler Cartan-Hadamard manifolds in terms of geometric or operator-theoretic properties of measures.
A full description of the membership in the Schatten ideal $S_ p(A^2_ω)$ for $0<p<\infty$ of the Toeplitz operator acting on large weighted Bergman spaces is obtained.
We give a necessary and a sufficient condition for the boundedness of the Toeplitz product $T_FT_{G^*}$ on the vector valued Bergman space $L_a^2(\mathbb{C}^n)$, where $F$ and $G$ are matrix symbols with scalar valued Bergman space entries. The results generalize existing results in the scalar valued Bergman space case. We also characterize boundedness and invertibility of Toeplitz products $T_FT_{G^*}$ in terms of the Berezin transform, generalizing results found by Zheng and Stroethoff for the scalar valued Bergman space.
Using the approach based on sesquilinear forms, we introduce Toeplitz operator in the analytic Bergman space on the upper half-plane with strongly singular symbols, derivatives of measures. Conditions for boundedness and compactness of such operators are found. A procedure of reduction of Toeplitz operators in Bergman spaces of polyanalytic functions to operators with singular symbols in the analytic Bergman space by means of the creation-annihilation structure is elaborated, which leads to the description of the properties of the former operators
On the Localization of the Bergman Kernel and applications to Toeplitz theory
研究Bergman核在大线丛和Radon测度下的局部化现象，确认Zelditch猜想，并展示Toeplitz算子在组合下构成代数，其谱在Bernstein-Markov测度下均匀分布。
Some new characterizations on Carleson measures for weighted Bergman spaces on the unit ball involving product of functions are obtained. For these we characterize bounded and compact Toeplitz operators between weighted Bergman spaces. The above results are applied to characterize bounded and compact extended Cesàro operators and pointwise multiplication operators. The results are new even in the case of the unit disk.
The boundedness and compactness of Toeplitz operator from $A_ω^p$ to $A_ω^q$ with doubling weights $ω$ are studied in this paper. The characterizations of Schatten class Toeplitz operators and Volterra operators on $A_ω^2$ are also investigated.
We find a concrete integral formula for the class of generalized Toeplitz operators $T_a$ in Bergman spaces $A^p$, $1<p<\infty$, studied in an earlier work by the authors. The result is extended to little Hankel operators. We give an example of an $L^2$-symbol $a$ such that $T_{|a|} $ fails to be bounded in $A^2$, although $T_a : A^2 \to A^2$ is seen to be bounded by using the generalized definition. We also confirm that the generalized definition coincides with the classical one whenever the latter makes sense.
Toeplitz operators in Bergman space induced by radial measures
本文研究单位圆盘上径向Carleson--Bergman测度诱导的Bergman空间中的Toeplitz算子，证明其对角性并计算特征值序列与Berezin变换，给出测度刻画，证明特征值序列的Lipschitz连续性，并描述生成的交换C*-代数。
Toeplitz Operators and Berezin-type Operators on Different Bergman Spaces
本文研究管状域上不同加权Bergman空间之间Toeplitz算子和Berezin型算子的有界性与紧性，建立其与Carleson测度的联系并给出刻画。
The spectral structure of Toeplitz operators with polynomial symbols on the harmonic Bergman space
本文利用对称多项式理论，证明了调和Bergman空间上解析多项式符号的Toeplitz算子谱为闭曲线与有限个孤立特征值之并，从而验证了Weyl定理。
We study the boundedness and compactness of the generalized Volterra integral operator on weighted Bergman spaces with doubling weights on the unit disk. A generalized Toeplitz operator is defined and the boundedness, compactness and Schatten class of this operator are investigated on the Hilbert weighted Bergman space. As an application, Schatten class membership of generalized Volterra integral operators are also characterized. Finally, we also get the characterizations of Schatten class membership of generalized Toeplitz operator and generalized Volterra integral operators on the Hardy spac…
We present and study commutative Banach algebras generated by Toeplitz operators with generalized quasi-radial pseudo-homogeneous symbols acting on the Bergman space over the unit ball. We develop the Gelfand theory of these algebras and give some structural information about them. In particular, we provide a description of the radical of these algebras. This paper generalizes and completes the results from previous works related to Toeplitz operators with quasi-radial quasi-homogeneous symbols.
We study positive Toeplitz operators on the Bergman spaces having the fast decreasing weight functions in a certain class. We give the characterizations for the boundedness and compactness of Toeplitz operators in terms of their Berezin transforms and averaging functions. In particular, we show the equivalent conditions in order that Toeplitz operators on $A^2(ω)$ belong to the Schatten class $S_p$, $0<p<\infty$.
Commuting Slant Toeplitz Operators on the Bergman Space
本文研究了开单位圆盘Bergman空间上的斜Toeplitz算子，证明了两个此类算子可交换的充要条件是两者的有界解析函数相等。
In this paper, we give a universal description of the boundedness and compactness of Toeplitz operator $\mathcal{T}_μ^ω$ between Bergman spaces $A_η^p$ and $A_\upsilon^q$ when $μ$ is a positive Borel measure, $1<p,q<\infty$ and $ω,η,\upsilon$ are regular weights. By using Khinchin's inequality and Kahane's inequality, we get a new characterization of the Carleson measure for Bergman spaces induced by regular weights.
Weyl type theorems for hypercyclic, supercyclic, and Toeplitz operators
本文研究超循环与超循环算子的性质$(UW_E)$，探讨Bergman空间上Toeplitz算子在紧扰动下Weyl型定理的稳定性，并给出满足Weyl型定理的Toeplitz算子实例。
The definition of Toeplitz operators in the Bergman space $A^2(D)$ of square integrable analytic functions in the unit disk in the complex plane is extended in such way that it covers many cases where the traditional definition does not work. This includes, in particular, highly singular symbols such as measures, distributions, and certain hyper-functions.
For $α>-1$, let $A^2_α$ be the corresponding weighted Bergman space of the unit ball in $\mathbb{C}^n$. For a bounded measurable function $f$, let $T_f$ be the Toeplitz operator with symbol $f$ on $A^2_α$. This paper describes all the functions $f$ for which $T_f$ commutes with a given $T_g$, where $g(z)=z_{1}^{L_1}... z_{n}^{L_n}$ for strictly positive integers $L_1,..., L_n$, or $g(z)=|z_1|^{s_1}... |z_n|^{s_n}h(|z|)$ for non-negative real numbers $s_1,..., s_n$ and a bounded measurable function $h$ on $[0,1)$.
If $ψ$ is analytic on the open unit disk $\mathbb{D}$ and $φ$ is an analytic self-map of $\mathbb{D}$, the weighted composition operator $C_{ψ,φ}$ is defined by $C_{ψ,φ}f(z)=ψ(z)f (φ(z))$, when $f$ is analytic on $\mathbb{D}$. In this paper, we study normal, cohyponormal, hyponormal and normaloid weighted composition operators on the Hardy and weighted Bergman spaces. First, for some weighted Hardy spaces $H^{2}(β)$, we prove that if $C_{ψ,φ}$ is cohyponormal on $H^{2}(β)$, then $ψ$ never vanishes on $\mathbb{D}$ and $φ$ is univalent, when $ψ\not \equiv 0$ and $φ$ is not a constant function. M…
Toeplitz Operators on Weighted Bergman Spaces over Tubular Domains
本文利用管状域上的 Carlson 测度，研究了管状域上加权 Bergman 空间上 Toeplitz 算子有界性与紧性的充要条件，并给出了相关的 Carlson 测度结果。
In this paper, we show that on the weighted Bergman space of the unit disk the essential norm of a noncompact Hankel operator equals its distance to the set of compact Hankel operators and is realized by infinitely many compact Hankel operators, which is analogous to the theorem of Axler, Berg, Jewell and Shields on the Hardy space in Axler et al. (1979); moreover, the distance is realized by infinitely many compact Hankel operators with symbols continuous on the closure of the unit disk and vanishing on the unit circle.
Zero Products of Toeplitz operators on the Hardy and Bergman spaces over an annulus
本文研究环域上Hardy与Bergman空间中Toeplitz算子的零积问题，在符号傅里叶展开条件下证明无零因子，并借助Abrahamse约化定理刻画紧Hankel算子，得出零积结果。
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