The Berezin Transform on the Toeplitz Algebra

Berezin-toeplitz quantization and complex weyl quantization of the torus t${}^2$

In this paper, we give a correspondence between the Berezin-Toeplitz and the complex Weyl quantizations of the torus $ \mathbb{T}^2$. To achieve this, we use the correspondence between the Berezin-Toeplitz and the complex Weyl quantizations of the complex plane and a relation between the Berezin-Toeplitz quantization of a periodic symbol on the real phase space $\mathbb{R}^2$ and the Berezin-Toeplitz quantization of a symbol on the torus $ \mathbb{T}^2 $.

Berezin transform and Toeplitz operators on weighted Bergman spaces induced by regular weights

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 …

Berezin regularity of domains in C^n and the essential norms of Toeplitz operators

For the open unit disc $\mathbb{D}$ in the complex plane, it is well known that if $ϕ\in C(\overline{\mathbb{D}})$ then its Berezin transform $\widetildeϕ$ also belongs to $C(\overline{\mathbb{D}})$. We say that $\mathbb{D}$ is BC-regular. In this paper we study BC-regularity of some pseudoconvex domains in $\mathbb{C}^n$ and show that the boundary geometry plays an important role. We also establish a relationship between the essential norm of an operator in a natural Toeplitz subalgebra and its Berezin transform.

当Berezin变换无法检测紧性时：加权Bergman空间上具有$L^1$符号的Toeplitz算子

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。

Berezin-Toeplitz Quantization of compact Kaehler manifolds

Invited lecture at the XIV-th workshop on geometric methods in physics, Bialowieza, Poland, July 9-15, 1995. In this lecture results are reviewed obtained by the author together with Martin Bordemann and Eckhard Meinrenken on the Berezin-Toeplitz quantization of compact Kaehler manifolds. Using global Toeplitz operators, approximation results for the quantum operators are shown. From them it follows that the quantum operators have the correct classical limit. A star product deformation of the Poisson algebra is constructed.

正Berezin liminf不意味着径向Toeplitz算子在Bergman和Fock空间上的本质正性

Positive Berezin liminf does not imply essential positivity for radial Toeplitz operators on Bergman and Fock spaces

研究径向Toeplitz算子在Bergman和Fock空间上的本质正性是否可通过Berezin变换的渐进行为检测。构造了具有正Berezin liminf但本质谱包含负点的径向符号，推翻Perälä-Virtanen猜想。

Fock空间之间的核Toeplitz算子

Nuclear Toeplitz operators between Fock spaces

研究Borel测度符号的Toeplitz算子$T_\mu: F_\alpha^p \to F_\alpha^q$的核性，给出了$q\leq p$时基于Berezin变换的充要条件，并揭示了$p<q$时Berezin变换不足以完全刻画。

径向加权Bergman空间上通过Berezin变换的紧Toeplitz算子

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。

Berezin-Type Operators on the Cotangent Bundle of a Nilpotent Group

We define and study coherent states, a Berezin-Toeplitz quantization and covariant symbols on the product between a connected simply connected nilpotent Lie group and the dual of its Lie algebra. The starting point is a Weyl system codifying the natural Canonical Commutation Relations of the system. The formalism is meant to complement the quantization of the cotangent bundle by pseudo-differential operators, to which it is connected in an explicit way. Some extensions are indicated, concerning $τ$-quantizations and variable magnetic fields.

Localization and Toeplitz Operators on Polyanalytic Fock Spaces

The well know conjecture of {\it Coburn} [{\it L.A. Coburn, {On the Berezin-Toeplitz calculus}, Proc. Amer. Math. Soc. 129 (2001) 3331-3338.}] proved by {\it Lo} [{\it M-L. Lo, {The Bargmann Transform and Windowed Fourier Transform}, Integr. equ. oper. theory, 27 (2007), 397-412.}] and {\it Englis} [{\it M. Engli$\check{s}$, Toeplitz Operators and Localization Operators, Trans. Am. Math Society 361 (2009) 1039-1052.}] states that any {\it Gabor-Daubechies} operator with window $ψ$ and symbol ${\bf a}(x,ω)$ quantized on the phase space by a {\it Berezin-Toeplitz} operator with window $Ψ$ and sy…

$H^2(\u27e8\ube40^n)$上通过Berezin变换的Toeplitz代数与符号映射

Toeplitz algebra and Symbol map via Berezin transform on $H^2(\mathbb{D}^n)$

本文利用Berezin变换研究开单位多圆盘上Hardy空间算子的类Douglas定理，得到一类更大的$C^*$-代数仍满足符号映射类似结论。

On convergence of the Berezin transforms

We prove approximation results about sequences of Berezin transforms of finite sums of finite product of Toeplitz operators (and bounded linear maps, in general) in the spirit of Ramadanov and Skwarczynski theorems that are about convergence of Bergman kernels.

Time-Frequency Localization Operators and a Berezin Transform

Time-frequency localization operators are a quantization procedure that maps symbols on $R^{2d}$ to operators and depends on two window functions. We study the range of this quantization and its dependence on the window functions. If the short-time Fourier transform of the windows does not have any zero, then the range is dense in the Schatten $p$-classes. The main tool is new version of the Berezin transform associated to operators on $L^2(R^d)$. Although some results are analogous to results about Toeplitz operators on spaces of holomorphic functions, the absence of a complex structure requi…

Toeplitz operators on the Fock space via the Fourier transform

In sprite by Berger-Coburn theorems and their conjecture in \cite{Coburn1994}, we use the Fourier transform to decompose $ T_{g}$ as an infinite sum of Toeplitz operators with symbols which have compact support in the frequency domain. As a consequence, we obtain a sufficient condition for $ T_{g}$ to be bounded in terms of the Carleson measure conditions defined by the heat transform of the symbol $g$. Moreover the decomposition of a Toeplitz operator leads us to get easily understanding that for a bounded function $g$, if its Berezin transform vanishes at infinity, then the Toeplitz operator…

Berger-Coburn theorem, localized operators, and the Toeplitz algebra

We give a simplified proof of the Berger-Coburn theorem on the boundedness of Toeplitz operators and extend this theorem to the setting of $p$-Fock spaces $(1\leq p \leq \infty)$. We present an overview of recent results by various authors on the compactness characterization via the Berezin transform for certain operators acting on the Fock space. Based on these results we present three new characterizations of the Toeplitz $C^*$ algebra generated by Toeplitz operators with bounded symbols.

Localized Frames and Compactness

We introduce the concept of weak-localization for generalized frames and use this concept to define a class of weakly localized operators. This class contains many important operators, including: Short Time Fourier Transform multipliers, Calderon-Toeplitz operators, Toeplitz operators on various functions spaces, Anti-Wick operators, and many others. In this paper, we study the boundedness and compactness of weakly localized operators. In particular, we provide a characterization of compactness for weakly localized operators in terms of the behavior of their Berezin transform.

Bergman 空间与 Fock 空间上的 Douglas 问题

The Douglas question on the Bergman and Fock spaces

本文研究 Bergman 空间与 Fock 空间上由正测度诱导的 Toeplitz 算子的可逆性，利用 Berezin 变换和反向 Carleson 条件给出若干充要条件，并证明两类空间上的 Douglas 问题在一般情形下答案均为否定。

Fixed points of the Berezin transform of polyanlytic Fock spaces

We study the fixed points of the Berezin transform in polyanalytic Fock spaces of $\mathbb{C}$. We show that an $L^p$ function, $p\in[1,+\infty]$, with respect to the Lebesgue measure is invariant under this transformation if and only if it is harmonic. From this we deduce that the only bounded fixed points of the Berezin transform of polyanalytic Fock spaces are constant functions.

广义Fock空间上的Berezin变换的乘积

Products of Berezin Transforms on Generalized Fock Spaces

该研究针对复空间ℂᵈ上广义Fock空间的Berezin变换乘积，对m≠2建立了参数唯一性结论，还刻画了Berezin变换的可交换性及乘积性质。

Bergman Toeplitz 算子的可逆性

Invertibility of Bergman Toeplitz operators

本文研究了 Bergman 空间上 Toeplitz 算子的可逆性问题，证明了符号的 Berezin 变换的可逆性是 Toeplitz 算子可逆的充要条件，并给出了特定函数形式下基于 Berezin 变换下确界的等价判据。

Schatten class and Berezin transform of quaternionic linear operators

In this paper we introduce the Schatten class of operators and the Berezin transform of operators in the quaternionic setting. The first topic is of great importance in operator theory but it is also necessary to study the second one because we need the notion of trace class operators, which is a particular case of the Schatten class. Regarding the Berezin transform, we give the general definition and properties. Then we concentrate on the setting of weighted Bergman spaces of slice hyperholomorphic functions. Our results are based on the S-spectrum for quaternionic operators, which is the not…

Bergman 空间上 Toeplitz 算子的紧性

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。

Compactness characterization of operators in the Toeplitz algebra of the Fock space $F_α^p$

For $1 < p < \infty$ let $\mathcal{T}_p ^α$ be the norm closure of the algebra generated by Toeplitz operators with bounded symbols acting on the standard weighted Fock space $F_α^p$. In this paper, we will show that an operator $A$ is compact on $F_α^p$ if and only if $A \in \mathcal{T}_p ^α$ and the Berezin transform $B_α(A)$ of $A$ vanishes at infinity.

Limit Operators, Compactness and Essential Spectra on Bounded Symmetric Domains

This paper is a follow-up to a recent article about the essential spectrum of Toeplitz operators acting on the Bergman space over the unit ball. As mentioned in the said article, some of the arguments can be carried over to the case of bounded symmetric domains and some cannot. The aim of this paper is to close the gaps to obtain comparable results for general bounded symmetric domains. In particular, we show that a Toeplitz operator on the Bergman space $A^p_ν$ is Fredholm if and only if all of its limit operators are invertible. Even more generally, we show that this is in fact true for all …

Toeplitz operators with analytic symbols

We provide asymptotic formulas for the Bergman projector and Berezin-Toeplitz operators on a compact K{ä}hler manifold. These objects depend on an integer N and we study, in the limit N $\rightarrow$ +$\infty$, situations in which one can control them up to an error O(e^{-cN}) for some c > 0. We develop a calculus of Toeplitz operators with real-analytic symbols, which applies to K{ä}hler man-ifolds with real-analytic metrics. In particular, we prove that the Bergman kernel is controlled up to O(e^{-cN}) on any real-analytic K{ä}hler manifold as N $\rightarrow$ +$\infty$. We also prove that To…

Range of Berezin Transform

Let $\ds dA=\frac{dxdy}π$ denote the normalized Lebesgue area measure on the unit disk $\disk$ and $u$, a summable function on $\disk$. $$B(u)(z)=\int_\disk u(ζ)\frac{(1-|z|^2)^2}{|1-ζ\oln z|^4}dA(ζ)$$ is called the Berezin transform of $u$. Ahern \cite{a} described all the possible triples $\{u,f,g\}$ for which $$B(u)(z)=f(z)\oln g(z)$$ where both $f,g$ are holomorphic in $\disk$. This result was crucial in solving a version of the zero product problem for Toeplitz operators on the Bergman space. The natural next question was to describe all functions in the range of Berezin Transform which a…

On regularity of the Berezin transform on smooth pseudoconvex domains

In this short note we improve some of recent results of Cuckovic and Sahutoglu concerning regularity of the Berezin transform for a class of smooth pseudoconvex domains.

Generalized Berezin Transform and Commutator Ideals

For a quasi-free module over a function algebra $A(Ω)$, we define an analogue of the Berezin transform and relate this to the quotient of the C*-algebra it generates modulo the commutator ideal.

Berezin transforms on noncommutative polydomains

This paper is an attempt to unify the multivariable operator model theory for ball-like domains and commutative polydiscs, and extend it to a more general class of noncommutative polydomains. An important role in our study is played by noncommutative Berezin transforms associated with the elements of the polydomain.

带垂直权重的半空间上的调和Berezin变换

Harmonic Berezin transform on half-space with vertical weights

该研究针对上半空间带垂直权重的调和Bergman空间，借助Siegel域全纯情形的已知结论，推导得到加权调和Bergman核与垂直坐标函数调和Berezin变换的完整渐近展开式。

Noncommutative Berezin transforms and model theory

We initiate the study of a class of noncommutative domains of n-tuples of bounded linear operators on a Hilbert space, which is generated by certain positivity conditions on polynomials in n noncommutative indeterminates. We obtain Fatou type results, functional calculi, and a model theory for n-tuples of operators in these domains. The main role in this study is played by a class of noncommutative Berezin transforms which is introduced in this paper. Our results extend to noncommutative varieties.

Berezin变换的$L^p$正则性的一个充分条件

A sufficient condition for $L^p$ regularity of the Berezin transform

本文证明了Berezin变换在$\nathbb{C}^n$的一大类域上具有$L^p$正则性，但在Hartogs三角形上缺乏$L^2$正则性。

Generalized Crofoot Transform and Matrix Valued Asymmetric Truncated Toeplitz Operators

Matrix valued asymmetric truncated Toeplitz operator are compression of multiplication operators acting between two model spaces. These are the generalization of matrix valued truncated Toeplitz operators. In this paper we use generalized Crofoot transform to characterize symbol of matrix valued asymmetric truncated Toeplitz operator equal to zero operator.

Relativistic Quantization and Improved Equation for a Free Relativistic Particle

Usually the only difference between relativistic quantization and standard one is that the Lagrangian of the system under consideration should be Lorentz invariant. The standard approaches are logically incomplete and produce solutions with unpleasant properties: negative-energy, superluminal propagation etc. We propose a two-projections scheme of (special) relativistic quantization. The first projection defines the quantization procedure (e.g. the Berezin-Toeplitz quantization). The second projection defines a casual structure of the relativistic system (e.g. the operator of multiplication by…

Berezin transforms on noncommutative varieties in polydomains

In this paper, we study noncommutative varieties in polydomains in $B(H)^n$. The goal is to understand the structure of these varieties, determine their elements and classify them up to unitary equivalence. Using noncommutative Berezin transforms associated with each variety, we develop an operator model theory and dilation theory for large classes of varieties in noncommutative polydomains. This includes various commutative cases which are close connected to the theory of holomorphic functions in several complex variables and algebraic geometry.

Fock型空间上Berezin变换的固定点

Fixed points of the Berezin transform on Fock-type spaces

本文研究了Fock型空间上Berezin变换的固定点，证明多项式固定点必须为调和多项式，除了可能的可数多个参数情况。

Amalgamated R-transform Theory on Toeplitz-Probability Spaces over Toeplitz Matricial Algebras

In this paper, we will consider a noncommutative probability space, $(T,E),$ over a Toeplitz matricial algebra $B=\QTR{cal}{C}^{N},$ for $N\in \QTR{Bbb}{N}$, induced by a (scalar-valued) noncommutative probability space, $(A,\QTR{cal}ϕ),$ with a suitable $B$-functional, $E:T\to B,$ defined by $ϕ.$ On this framework, we will observe amalgamated R-transform calculus with respect to a $B$-functional, $E.$ The technique and idea are came from those in [12] and Free Probability of type $B$ studied in [31].

On Function Theory in Quantum Disc: A q-Analogue of Berezin Transform

This work contains a proof of theorem 7.3 from math.QA/9808015. This theorem demonstrates the Berezin method to be applicable for producing a well known one-parameter deformation of the quantum disc.

Coherent states quantization of generalized bergman spaces on the unit ball of cn with a new formula for their associated berezin transforms

While dealing with a class of generalized Bergman spaces on the unit ball, we construct for each of these spaces a set of coherent states to apply a coherent states quantization method. This provides us with another way to recover the Berezin transforms attached to these spaces. Finally, a new formula representing these transforms a functions of the Laplace-Beltrami operator is established in terms ofWilson polynomials by using the Fourier-Helgason transform.

Guillemin Transform and Toeplitz Representations for Operators on Singular Manifolds

An approach to the construction of index formulas for elliptic operators on singular manifolds is suggested on the basis of K-theory of algebras and cyclic cohomology. The equivalence of Toeplitz and pseudodifferential quantizations, well known in the case of smooth closed manifolds, is extended to the case of manifolds with conical singularities. We describe a general construction that permits one, for a given Toeplitz quantization of a C^*-algebra, to obtain a new equivalent Toeplitz quantization provided that a resolution of the projection determining the original quantization is given.

Tensorial Function Theory: From Berezin transforms to Taylor's Taylor series and back

Let $H^{\infty}(E)$ be the Hardy algebra of a $W^{*}$-correspondence $E$ over a $W^{*}$-algebra $M$. Then the ultraweakly continuous completely contractive representations of $H^{\infty}(E)$ are parametrized by certain sets $\mathcal{AC}(σ)$ indexed by $NRep(M)$ - the normal *-representations $σ$ of $M$. Each set $\mathcal{AC}(σ)$ has analytic structure, and each element $F\in H^{\infty}(E)$ gives rise to an analytic operator-valued function $\hat{F}_σ$ on $\mathcal{AC}(σ)$ that we call the $σ$-Berezin transform of $F$. The sets ${\mathcal{AC}(σ)}_{σ\inΣ}$ and the family of functions ${\hat{F}…

The Bargmann transform on a broad family of Banach spaces, with applications to Toeplitz and pseudo-differential operators

We investigate mapping properties for the Bargmann transform on modulation spaces whose weights and their reciprocals are allowed to grow faster than exponentials. We prove that this transform is isometric and bijective from modulation spaces to convenient Lebesgue spaces of analytic functions. We use this to prove that such modulation spaces fulfill most of the continuity properties which are well-known when the weights are moderated. Finally we use the results to establish continuity properties of Toeplitz and pseudo-differential operators in the context of these modulation spaces.