Positivity of Toeplitz operators on harmonic Bergman space

In this paper, we study positive Toeplitz operators on the harmonic Bergman space via their Berezin transforms. We consider the Toeplitz operators with continuous harmonic symbols on the closed disk and show that the Toeplitz operator is positive if and only if its Berezin transform is nonnegative on the disk. On the other hand, we construct a function such that the Toeplitz operator with this function as the symbol is not positive but its Berezin transform is positive on the disk. We also consider the harmonic Bergman space on the upper half plane and prove that in this case the positive Toeplitz operators with continuous integrable harmonic symbols must be the zero operator.     

Berezin transform and Toeplitz operators on harmonic Bergman spaces

For an operator which is a finite sum of products of finitely many Toeplitz operators on the harmonic Bergman space over the half-space, we study the problem: Does the boundary vanishing property of the Berezin transform imply compactness? This is motivated by the Axler-Zheng theorem for analytic Bergman spaces, but the answer would not be yes in general, because the Berezin transform annihilates the commutator of any pair of Toeplitz operators. Nevertheless we show that the answer is yes for certain subclasses of operators. In order to do so, we first find a sufficient condition on general operators and use it to reduce the problem to whether the Berezin transform is one-to-one on related subclasses.     

Dirichlet空间上Toeplitz算子与Berezin型变换相关的一些性质

本文研究了单位圆盘D 的Dirichlet 空间上Toeplitz 算子和小Hankel 算子. 利用Berezin 型变换讨论了Toeplitz 算子的不变子空间问题, 具有Berezin 型符号的Toeplitz 算子的渐进可乘性以及Toeplitz 算子的Riccati 方程的可解性. 应用Berezin 变换得到了Toeplitz 算子和小Hankel 算子可逆的充分条件. 此外, 还利用Hankel 算子和Berezin 变换刻画了算子2T_uv-T_uT_v-T_vT_u 的紧性, 其中函数u,v ∈ L^2,1.     

Two-weight Lp- Lq Bounds for Positive Dyadic Operators: Unified Approach to p ≤ q and p > q

We characterize the bounded and compact generalized Volterra companion integral operators acting between the standard Fock spaces. We also obtain asymptotic estimates for the norm of these operators in terms of certain Berezin type integral transforms which involve some function theoretic properties of symbols inducing the operators. As a special case, we prove that there exist no nontrivial compact Volterra companion integral and multiplication operators on Fock spaces.     

Berezin versus Mellin

We study zero products of two Bergman space Toeplitz operators, where one symbol is harmonic. Our results point in the direction of the zero product problem having only a trivial solution. The techniques we use are based on a formula that connects the Berezin and Mellin transform.     

Toeplitz operators on Fock-Sobolev spaces with positive measure symbols

We discuss Toeplitz operators on Fock-Sobolev space with positive measure symbols.By FockCarleson measure,we obtain the characterizations for boundedness and compactness of Toeplitz operators.We also give some equivalent conditions of Schatten p-class properties of Toeplitz operators by Berezin transform.     

Localized Frames and Compactness

We introduce the concept of weak localization for continuous frames and use this concept to define a class of weakly localized operators. This class contains many important classes of operators, including: short time Fourier transform multipliers, Calderon–Toeplitz operators, Toeplitz operators on various functions spaces, Anti-Wick operators, some pseudodifferential operators, some Calderon–Zygmund 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 transforms.     

On the Berezin Symbol

We discuss some questions related to the Berezin symbols of bounded linear operators on Hilbert function spaces. Bibliography: 14 titles.     

The Berezin transform and Laplace-Beltrami operator

We study the Berezin transform of bounded operators on the Bergman space on a bounded symmetric domain Ω in Cn. The invariance of range of the Berezin transform with respect to G=Aut(Ω), the automorphism group of biholomorphic maps on Ω, is derived based on the general framework on invariant symbolic calculi on symmetric domains established by Arazy and Upmeier. Moreover we show that as a smooth bounded function, the Berezin transform of any bounded operator is also bounded under the action of the algebra of invariant differential operators generated by the Laplace-Beltrami operator on the unit disk and even on the unit ball of higher dimensions.     

有界多连通域的Schatten类Toeplitz算子

本文讨论了多连通域的Bergman空间上的以正测度为符号的Toeplitz算子.用符号测度的Berezin 变换和平均函数刻画了Toeplitz算子为Schatten类算子的充要条件.     

n维单位球面上的Toeplitz符号演算

探讨了C^n中单位球面S上Berezin变换和Toeplitz算子的性质，证明了由{Tφ,φ∈L^∞ （S）}所生成的C^＊-代数中算子T的符号恰好为单位球B上函数T（称为T的Berezin变换）的非切向边界值．此外，本文还得到了经典Toeplitz符号演算的有趣推广．     

Some results for operators on a model space

We investigate some problems for truncated Toeplitz operators. Namely, the solvability of the Riccati operator equation on the set of all truncated Toeplitz operators on the model space K _θ = H²ΘθH² is studied. We study in terms of Berezin symbols invertibility of model operators. We also prove some results for the Berezin number of the truncated Toeplitz operators. Moreover, we study some property for H²-functions in terms of noncyclicity of co-analytic Toeplitz operators and hypercyclicity of model operators.     

Schatten class and Berezin transform of quaternionic linear operators

In this paper, we introduce the Schatten class and the Berezin transform of quaternionic operators. The first topic is of great importance in operator theory, but it is also necessary to study the second one, which requires the notion of trace class operators, 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 of quaternionic operators, which is the notion of spectrum that appears in the quaternionic version of the spectral theorem and in the quaternionic S‐functional calculus. Copyright © 2016 John Wiley & Sons, Ltd.     

Positivity of Toeplitz operators via Berezin transform

In this paper, we study positive Toeplitz operators on the Bergman space via their Berezin transforms. Surprisingly we show that the positivity of a Toeplitz operator on the Bergman space is not completely determined by the positivity of the Berezin transform of its symbol. In fact, we show that even if the minimal value of the Berezin transform of a quadratic polynomial of |z|$|z|$ on the unit disk is positive, the Toeplitz operator with the function as the symbol may not be positive.     

The Berezin Transform and Radial Operators on the Bergman Space of the Unit Ball

In this paper, we investigate the connection between compactness of operators on the Bergman space and the boundary behaviour of the corresponding Berezin transform. We prove that for a class of operators that we call radial operators, an oscillation criterion and diagonal are sufficient conditions under which the compactness of an operator is equivalent to the vanishing of the Berezin transform on the unit sphere. We further study a special class of radial operators, i.e., Toeplitz operators with a radial L ¹(B _n ) symbol.     

高维加权Bergman空间上的Toeplitz算子

本文研究了高维加权Bergman空间Ap(Bn,dVpφ)(1＜p＜∞).上的Toeplitz算子.利用Toeplitz算子的Berezin变换,获得了Ap(Bn,dVp)(1＜p＜∞)上具有L∞(Bn)符号的Toeplitz算子的有限乘积的有限和是紧算子的一些等价刻画,推广了加权Bergman空间Ap(D,dmpφ)上的Toeplitz算子的有限乘积的有限和是紧的当且仅当它的Berezin变换在单位圆盘的边界消失为0的结论     

(e)-convergence and related problem

We answer negatively to a question of Zorboska (2003) [13], which is concerned to the boundary behavior of Berezin symbols of Bergman space operators. We also introduce the notions of (e)-summability of sequences and series of complex numbers, and study some of their properties. As a corollary, we obtain the classical Abel theorems of summability theory.     

Berezin symbol and invertibility of operators on the functional Hilbert spaces

We give in terms of reproducing kernel and Berezin symbol the sufficient conditions ensuring the invertibility of some linear bounded operators on some functional Hilbert spaces.     

加权Bergman空间上的紧算子

本文讨论了加权Ｂｅｒｇｍａｎ空间上的Ｔｏｅｐｌｉｔｚ算子,证明了Ｔｏｐｌｉｔｚ算子的有限乘积的有限和是紧的当且仅当它的Ｂｅｒｅｚｉｎ变换在边界上趋向于零．     

调和Bergman空间上的Toeplitz的零积及有限秩性质

本文研究了调和Bergman空间上的Toeplitz的Brown-Halmos型零乘积问题以及有限秩的相关性质,并得到了以解析和余解析符号的Toeplitz算子乘积的有限和的零积有限秩的等价条件.