Sheldon Axler, Zeljko Cuckovic, N. V. Rao
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.
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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.
We make a progress towards describing the commutants of Toeplitz operators with harmonic symbols on the Bergman space over the unit disk. Our work greatly generalizes several partial results in the field.
We make a progress towards describing the semi-commutants of Toeplitz operators on Fock-Sobolev spaces of nonnegative orders. We generalize the results in \cite{Bauer1,Qin}. For the certain symbol spaces, we obtain two Toeplitz operators can semi-commute only in the trivial cases, which is different from what is known for the classical Fock spaces. As an application, we consider the conjecture which was shown to be false for Fock space in \cite{MA}. The main results of this paper say that there is the fundamental difference between the geometries of Fock and Fock-Sobolev space.
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 $[0,1)$.
In this note we describe the commutant of the multiplication operator by a monomial in the Toeplitz algebra of a complete strongly pseudoconvex Reinhardt domain.
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.
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)
In the setting of the Fock space over the complex plane, Bauer and Lee have recently characterized commutants of Toeplitz operators with radial symbols, under the assumption that symbols have at most polynomial growth at infinity. Their characterization states: If one of the symbols of two commuting Toeplitz operators is nonconstant and radial, then the other must be also radial. We extend this result to the Fock-Sobolev spaces.
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.
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…
We provide a characterization of the commutant of analytic Toeplitz operators $T_B$ induced by finite Blachke products $B$ acting on weighted Bergman spaces which, as a particular instance, yields the case $B(z)=z^n$ on the Bergman space solved recently by by Abkar, Cao and Zhu. Moreover, it extends previous results by Cowen and Wahl in this context and applies to other Banach spaces of analytic functions such as Hardy spaces $H^p$ for $1<p<\infty$. Finally, we apply this approach to study the reducing subspaces of $T_{B}$ in weighted Bergman spaces.
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.
In this paper we study the Toeplitz algebra, which is generated by Toeplitz operators with bounded symbols on the Fock space $F^p_α$. We show that the Toeplitz algebra coincides with each of the algebras generated by band-dominated, sufficiently localized and weakly localized operators, respectively. Moreover, we determine its essential commutant and its essential bicommutant. For $p = 2$ these results were obtained recently by Xia. However, Xia's ideas are mostly connected to Hilbert space theory and methods which are not applicable for $p \neq 2$. Instead, we use a recent result of Fulsche t…
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 $1<p<\infty$.
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.
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。
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 $A_φ^{2}$.
Minimal commutant and double commutant property for analytic Toeplitz operators
该数学研究给出了解析Toeplitz算子换位子极小性的等价刻画，并将其应用于判定整函数及Thomson-Cowen类符号对应的算子双换位子极小性。
When is the collection of $\mathsf S$-Toeplitz operators with respect to a tuple of commuting bounded operators $\mathsf S= (S_1, S_2, \ldots , S_{d-1}, P)$, which has the symmetrized polydisc as a spectral set, non-trivial? The answer is in terms of powers of $P$ as well as in terms of a unitary extension. En route, Brown-Halmos relations are investigated. A commutant lifting theorem is established. Finally, we establish a general result connecting the $C^*$-algebra generated by the commutant of $\mathsf S$ and the commutant of its unitary extension $\mathsf R$.
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.
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…
We prove an approximation theorem on a class of domains in $\mathbb{C}^n$ on which the $\overline{\partial}$-problem is solvable in $L^{\infty}$. Furthermore, as a corollary, we obtain a version of the Axler-Čučković-Rao Theorem in higher dimensions.
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.
Commutants of a certain class of Toeplitz operators
针对解析 Bergman 空间中 Toeplitz 算子的换位子刻画问题，本文完整描述了与拟齐次函数加有界解析函数符号的 Toeplitz 算子交换、且符号具截断极分解的有界 Toeplitz 算子。
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。
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 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.
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.
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。
A geometric condition for the invertibility of Toeplitz operators on the Bergman space
针对Bergman空间上Toeplitz算子可逆性及Douglas公开问题，提出符号映射的全新几何条件（放宽标准正性条件），证明对应算子可逆等价于|φ|的Berezin变换在L^∞中可逆，并刻画了一类一般调和多项式对应Toeplitz算子的可逆性，给出相关结果与示例。
In this article we study commutant lifting, more generally intertwining lifting, for different reproducing kernel Hilbert spaces over two domains in $\mathbb{C}^n$, namely the unit ball and the unit polydisc. The reproducing kernel Hilbert spaces we consider are mainly weighted Bergman spaces. Our commutant lifting results are explicit in nature and that is why these results are new even in one variable $(n=1)$ set up.
On the essential spectra of Toeplitz operators on Bergman spaces of the bi-disc
本文研究双圆盘 Bergman 空间上符号为连续函数的 Toeplitz 算子的本质谱确定问题，给出了相应本质谱的刻画。
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。
Let $\mathfrak{T}$ denote the full Toeplitz algebra on the Bergman space of the unit ball $\mathbb{B}_n.$ For each subset $G$ of $L^{\infty},$ let $\mathfrak{CI}(G)$ denote the closed two-sided ideal of $\mathfrak{T}$ generated by all $T_fT_g-T_gT_f$ with $f,g\in G.$ It is known that $\mathfrak{CI}(C(\bar{\mathbb{B}}_n))=\mathcal{K}$ - the ideal of compact operators and $\mathfrak{CI}(C(\mathbb{B}_n))=\mathfrak{T}.$ Despite these ``extremal cases'', $\mathfrak{T}$ does contain other non-trivial ideals. This paper gives a construction of a class of subsets $G$ of $L^{\infty}$ so that $\mathcal{…
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
In this paper, using the group-like property of local inverses of a finite Blaschke product $ϕ$, we will show that the largest $C^*$-algebra in the commutant of the multiplication operator $M_ϕ$ by $ϕ$ on the Bergman space is finite dimensional, and its dimension equals the number of connected components of the Riemann surface of $ϕ^{-1}\circϕ$ over the unit disk. If the order of the Blaschke product $ϕ$ is less than or equal to eight, then every $C^*$-algebra contained in the commutant of $M_ϕ$ is abelian and hence the number of minimal reducing subspaces of $M_ϕ$ equals the number of connect…
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.
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.
Localizations and Essential Commutant of Toeplitz Algebra on Polydisk
该研究证明多圆盘Bergman空间上有界符号Toeplitz算子族的范数闭包等价于Toeplitz代数及算子类$\u212b$，并刻画了Toeplitz代数的本质换位子，验证了其在Calkin代数中的双重换位子关系。
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.
Spectral properties of Toeplitz operators with harmonic function symbols on the Bergman space
本文研究了贝拉曼空间上具有调和函数符号的托埃利茨算子的谱性质，揭示了其谱结构和Fredholm指数特性。
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.
We study the ranks of commutators and generalized semicommutators of Toeplitz operators with quasihomogeneous symbols on both the harmonic Bergman space and the Bergman Space. In particular, when one of quasihomogeneous symbols is the the form of $e^{ikθ}r^{m}$, we first obtain specific sufficient and necessary conditions for commutators and generalized semicommutators to be finite rank. Then we make further efforts to determine the range of each finite rank commutator and generalized semicommutators, and consequently the explicit canonical form and the rank are obtained. Thus, the finite rank…
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.
The essential norm of Toeplitz operators between Bergman spaces induced by doubling weights
本文研究加倍权诱导的Bergman空间间Toeplitz算子的本质范数，利用Carleson块和径向权结构给出精确估计，并解决有界性到紧性的过渡问题，推广了经典结果。
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.
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
Absolutely Summing Toeplitz operators on Bergman spaces in the unit ball of $\mathbb{C}^n$
本文研究了复数n维单位球上Bergman空间中Toeplitz算子的绝对可和性，并给出了其范数等价条件，同时扩展了Carleson嵌入的相关结果。
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 prove some characterizations of Schatten class Toeplitz operators on Bergman spaces of tube domains over symmetric cones for small exponents.
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。
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…
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