Grigori Rozenblum
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
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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.
Schatten-Herz class Toeplitz operators on weighted Bergman spaces induced by doubling weights are investigated in this paper.
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.
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.
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
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.
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…
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.
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.
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。
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。
Toeplitz operators on the weighted Bergman spaces of quotient domains
利用有限伪反射群的不变理论与表示理论，建立商域上加权Bergman空间Toeplitz算子的恒等式，并应用于广义零积问题与交换对刻画，在对称化多圆盘等特殊商域获得精细结果。
Toeplitz operators and weighted composition operators on variable exponent Bergman spaces
本文将Choe等人关于加权复合算子之差有界性与紧性的Carleson测度刻画推广到单位球上的变指数Bergman空间，并给出更简单的新证明，同时刻画了Toeplitz算子的有界性与紧性。
Hypercyclic Bergman-Toeplitz operators with some harmonic symbols
本文研究Bergman空间上符号为$a\ar{z}+p(z)$的Toeplitz算子的动力学性质，给出了超循环等性质的充要条件，并完整刻画了调和线性多项式符号的情形，还发现同一符号在不同空间上超循环行为可不同。
We prove some characterizations of Schatten class Toeplitz operators on Bergman spaces of tube domains over symmetric cones for small exponents.
In this paper we study mapping properties of Toeplitz-like operators on weighted Bergman spaces of bounded strongly pseudconvex domains in $\mathbb{C}^n$. In particular we prove that a Toeplitz operator built using as kernel a weighted Bergman kernel of weight $β$ and integrating against a measure $μ$ maps continuously (when $β$ is large enough) a weighted Bergman space $A^{p_1}_{α_1}(D)$ into a weighted Bergman space $A^{p_2}_{α_2}(D)$ if and only if $μ$ is a $(λ,γ)$-skew Carleson measure, where $λ=1+\frac{1}{p_1}-\frac{1}{p_2}$ and $γ=\frac{1}λ\left(β+\frac{α_1}{p_1}-\frac{α_2}{p_2}\right)$.…
Toeplitz operators via Carleson measures On $β$-modified Bergman Spaces
本文研究了 $β$-修正 Bergman 空间上的 Bergman 投影性质与 Toeplitz 算子紧致的充要条件，并通过引入等价于经典 Bergman-Poincaré 度量的新度量，利用 Carleson 测度刻画了 Toeplitz 算子。
On the Berezin range of Toeplitz and weighted composition operators on weighted Bergman spaces
本文刻画了加权Bergman空间上调和符号Toeplitz算子的Berezin值域，引入一类新的加权复合算子并确定其Berezin值域与Berezin数，同时研究了复合算子Berezin值域的凸性及原点与闭包的关系。
We characterize the Schatten class Toeplitz operators induced by a positive Borel measure on the unit disc and the reproducing kernel of the Bergman space $A^2_ω$, where $ω$ is a radial weight satisfying the doubling property $\int_r^1ω(s)\,ds\le C\int_{\frac{1+r}{2}}^1ω(s)\,ds$. By using this, we describe the Schatten class composition operators. We also discuss basic properties of composition operators acting from $A^p_ω$ to $A^q_v$.
For $-1<α<\infty$, let $ω_α(z)=(1+α)(1-|z|^2)^α$ be the standard weight on the unit disk. In this note, we provide descriptions of the boundedness and compactness for the Toeplitz operators $T_{μ,β}$ between distinct weighted Bergman spaces $L_{a}^{p}(ω_α)$ and $L_{a}^{q}(ω_β)$ when $0<p\leq1$, $q=1$, $-1<α,β<\infty$ and $0<p\leq 1<q<\infty, -1<β\leqα<\infty$, respectively. Our results can be viewed as extensions of Pau and Zhao's work in \cite{Pau}. Moreover, partial of main results are new even in the unweighted settings.
Inspired by our previous work on the boundedness of Toeplitz operators, we introduce weak BMO and VMO type conditions, denoted by BWMO and VWMO, respectively, for functions on the open unit disc of the complex plane. We show that the average function of a function $f$ in BWMO is boundedly oscillating, and the analogous result holds for $f$ in VWMO. The result is applied for generalizations of known results on the essential spectra and norms of Toeplitz operators. Finally, we provide examples of functions satisfying the VWMO condition which are not in the classical VMO or even in BMO.
Given a regular weight $ω$ and a positive Borel measure $μ$ on the unit disc $\mathbb{D}$, the Toeplitz operator associated with $μ$ is $$ \mathcal{T}_μ(f)(z)=\int_{\mathbb{D}} f(ζ)\bar{B_z^ω(ζ)}\,dμ(ζ), $$ where $B^ω_{z}$ are the reproducing kernels of the weighted Bergman space $A^2_ω$. We describe bounded and compact Toeplitz operators $\mathcal{T}_μ:A^p_ω\to A^q_ω$, $1<q,p<\infty$, in terms of Carleson measures and the Berezin transform $$ \widetilde{\mathcal{T}_μ}(z)=\frac{\langle\mathcal{T}_μ(B^ω_{z}), B^ω_{z} \rangle_{A^2_ω}}{\|B_z^ω\|^2_{A^2_ω}}. $$ We also characterize Schatten class …
Suppose that $ω$ is a radial weight on the unit disk that satisfies both forward and reverse doubling conditions. Using Carleson measures and $T1$-type conditions, we obtain necessary and sufficient conditions of the positive Borel measure $μ$ such that the Toeplitz operator $T_{μ,ω}:L^p_a(ω)\to L_a^1(ω)$ is bounded and compact for $0<p\leq 1$. In addition, we obtain a bump condition for the bounded Toeplitz operators with $L^1(ω)$ symbol on $L^1_a(ω)$. This generalizes a result of Zhu in \cite{zhu1989}.
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 study the Bergman-Toeplitz operator $T_ψ$ induced by $ψ(w) = K_Ω^{-α}(w,w)d_Ω^β(w)$ with $α, β\geq 0$ acting from a weighted $L^p$-space $L_a^p(Ω)$ to another one $L_a^q(Ω)$ on a large class of pseudoconvex domains of finite type. In the case $1 < p \leq q < \infty$, the following results are established: \\ - Necessary and sufficient conditions for boundedness, which generalize the recent results obtained by Khanh, Liu and Thuc.\\ - Upper and lower estimates for essential norm, in particular, a criterion for compactness.\\ - A characterization of Schatten class membership of …
We prove that for certain classes of pseudoconvex domains of finite type, the Bergman-Toeplitz operator $T_ψ$ with symbol $ψ=K^{-α}$ maps from $L^{p}$ to $L^{q}$ continuously with $1< p\le q<\infty$ if and only if $α\ge\frac{1}{p}-\frac{1}{q}$, where $K$ is the Bergman kernel on diagonal. This work generalises the results on strongly pseudoconvex domains by Čučković and McNeal, and Abeta, Raissy and Saracco.
A Szegö limit theorem for a class of Toeplitz operators on the Bergman space of the unit ball with singular symbols
本文针对单位球加权 Bergman 空间上符号支撑于迷向或余迷向子流形且关于 Lebesgue 测度奇异的一类 Toeplitz 算子，建立了 Szegö 极限定理，刻画了参数趋于无穷时的渐近行为。
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)
For a smoothly bounded strictly pseudoconvex domain, we describe the boundary singularity of weighted Bergman kernels with respect to weights behaving like a power (possibly fractional) of a defining function, and, more generally, of the reproducing kernels of Sobolev spaces of holomorphic functions of any real order. This generalizes the classical result of Fefferman for the unweighted Bergman kernel. Finally, we also exhibit a holomorphic continuation of the kernels with respect to the Sobolev parameter to the entire complex plane. Our main tool are the generalized Toeplitz operators of Bout…
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。
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…
Toeplitz and Hankel operators on Bergman spaces with doubling weights
本文研究倍权Bergman空间上Toeplitz与Hankel算子的有界性和紧性，通过Berezin变换刻画有界性，并给出共解析符号情形下的充要条件。
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 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.
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.
Bergman Projections, Kernel $p$-Norm Estimates, and Toeplitz Operators with Békollé and Bonami weights
本文针对加权贝科勒 - 博纳米伯格曼空间间的托普利兹算子，通过建立核p-范数估计来刻画其有界和紧性，借助广义贝雷津变换等描述性质，引入权重条件确保加权伯格曼投影有界，将相关结果从希尔伯特空间推广到巴拿赫空间。
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 $A_φ^{2}$.
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.
Compactness of Toeplitz Operators on the Bergman Space
本文用符号的局部平均值刻画 Bergman 空间上 Toeplitz 算子的紧性，证明紧性等价于 Bergman 圆盘平均值趋于零，并给出 Carleson 盒子平均的等价判据及反例。
When the Berezin transform fails to detect compactness: Toeplitz operators with $L^1$ symbols on weighted Bergman spaces
该研究针对加权Bergman空间，构造了具有$L^1$符号的Toeplitz算子，其Berezin变换在边界消失但算子非紧，回答了Bauer与Isralowitz的问题，还在无权重圆盘上验证了相关性质并确定Zorboska假设的最优性。
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 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.
We construct explicit examples of non-trivial nilpotent Toeplitz operators on Bergman spaces of certain Reinhardt domains in $\mathbb{C}^2$.
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…
Compact Toeplitz operators via the Berezin transform on radial weighted Bergman spaces
该研究针对径向$\widehat{\mathcal{D}}$-加权Bergman空间，证明了Toeplitz算子紧性等价于其Berezin变换在边界处趋于零，同时构造反例说明该刻画不适用于Toeplitz算子乘积及对应代数。
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