Nihat Gokhan Gogus, Sonmez Sahutoglu
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.
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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.
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。
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。
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。
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 transforms on modified Bergman spaces
本文研究了修正Bergman空间上Berezin变换的连续性，并基于Bergman-Poincaré度量建立了Lipschitz估计。
We study the reproducing kernel for weighted polynomial Bergman spaces and consider applications to the Berezin transform. Some of our results have applications in random matrix theory, a topic which we discuss in a separate (companion) paper.
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…
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.
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。
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。
In this lecture results on the Berezin-Toeplitz quantization of arbitrary compact quantizable Kaehler manifolds are presented. These results are obtained in joint work with M. Bordemann and E. Meinrenken. The existence of the Berezin-Toeplitz deformation quantization is also covered. Recent results obtained in joint work with A. Karabegov on the asymptotic expansion of the Berezin transform for arbitrary quantizable compact Kaehler manifolds are explained. As an application the asymptotic expansion of the Fubini-Study fundamental form under the coherent state embedding is considered. Some comm…
On the Commutativity of the Berezin Transform
本文研究加权Fock空间上Berezin变换的交换性，证明对所有有界函数及正参数，交换关系成立当且仅当参数$m=2$。
We consider a weighted Berezin transform: $$ B_α : L^{\infty} (\mathbb{B}^n) \to \ \mathcal{B},\quad α>-1,$$ defined, for $f \in L^{\infty} \left( \mathbb{B}^n \right)$ and $z \in \mathbb{B}^n$, by $$(B_αf) (z) = c_α\int_{\mathbb{B}^n} \frac{\left( 1-|z|^2 \right)^{n+1}}{|1 - \langle z, w \rangle|^{2n+2}} f(w) \left( 1-|w|^2 \right)^α\ d v(w),$$ where $c_α = \frac{Γ(α+n+1)}{Γ(α+1)π^n}$ , $v$ is the Lebesque measure and $\mathcal{B}$ is a Bloch-type space. We prove that $B_α$ is bounded iff $α>0$ and give the exact semi-norm of $ B_α$ for $0\leqα\leq 2n+3.$
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}…
We consider a bounded radial function f. B. Korenblum and K.E.Zhu give a case where we have equality between the limit near the boundary of the unit disc of the Berezin transform and the limit of the normalized Mellin coefficient when one of them is 0. We previously describe the case where one limit point has modulus infinite norm of f. We also use the mean values of f near 1. The aim of this article is to show that between these two extreme cases, we can have distinct limits. In the same time, we also study the sets of limit points of these quantities.
While dealing with a class of generalized Bargmann spaces, we rederive their reproducing kernels from the knowledge of an orthonormal basis by using an addition formula for Laguerre polynomials involving the disk polynomials. 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, two new formulae representing these transforms as functions of the Euclidean Laplacian are established and a possible physics direction for the application of s…
$L^p$-boundedness of Berezin transforms on generalized Hartogs triangles
研究有理广义Hartogs三角形上Berezin变换的$L^p$有界性，证明了其有界当且仅当$p>m+nl$。
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。
This work presents proofs of the main results of (math.QA/9808015), except those on q-Berezin transform to appear in a subsequent work. The notation and the results of (math.QA/9808037) and (math.QA/9808047) are used.
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。
Compactness and the essential norm on Bergman spaces of the Siegel upper half-space
本文利用Berezin变换刻画紧性，得到西格尔上半空间伯格曼空间上Toeplitz代数中算子的本质范数估计。
A. Poltotaski proved an analog of Carleson's Theorem on almost everywhere convergence of Fourier series for a version of the non-linear Fourier transform. We aim to present his proof in full detail and elaborate on the ideas behind each step.
Pointwise convergence of the non-linear Fourier transform
本文研究了非线性傅里叶变换的逐点收敛性，通过分析Dirac系统共振，证明了其与Carleson定理的类比结果。
Convergence from the discrete to the continuous non-linear Fourier transform
该研究借助复函数论与谱问题的关联新视角，探究离散到连续非线性傅里叶变换的收敛性，指出微分算子谱问题视角下的相关结果与离散情形的收敛问题具有相似性。
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