We give a survey on the Berezin transform and its applications in operator theory. The focus is on the Bergman space of the unit disk and the Fock space of the complex plane. The Berezin transform is most effective an...We give a survey on the Berezin transform and its applications in operator theory. The focus is on the Bergman space of the unit disk and the Fock space of the complex plane. The Berezin transform is most effective and most successful in the study of Hankel and Toepltiz operators.展开更多
In this paper, we analyze a class of bounded radial operators on the weighted Bergman space A2α(Bn, d Vα), we get that these kinds of operators are diagonal with respect to the standard orthonomal basis. We also inv...In this paper, we analyze a class of bounded radial operators on the weighted Bergman space A2α(Bn, d Vα), we get that these kinds of operators are diagonal with respect to the standard orthonomal basis. We also investigate the connection between compactness of operators and the boundary behaviour of the corresponding Berezin transform. We further study a special class of radial operators, i.e., Toeplitz operators with a radial L1 symbol.展开更多
Abstract This paper deals with the relationship between the positivity of the Fock Toeplitz operators and their Berezin transforms. The author considers the special case of the bounded radial function φ（z）=α ＋ be...Abstract This paper deals with the relationship between the positivity of the Fock Toeplitz operators and their Berezin transforms. The author considers the special case of the bounded radial function φ（z）=α ＋ be-α｜z｜2 ＋ ce-β｜z｜2, where a, b, c are real numbers and α,β are positive numbers. For this type of φ, one can choose these parameters such that the Berezin transform of is a nonnegative function on the complex plane, but the corresponding Toeplitz operator Tφ is not positive on the Fock space.展开更多
There is a singular integral operators Sj on the Fock space F2(C),which originated from the unitarily equivalent version of the Hilbert transform on L2(R).In this paper,we give an analytic characterization of function...There is a singular integral operators Sj on the Fock space F2(C),which originated from the unitarily equivalent version of the Hilbert transform on L2(R).In this paper,we give an analytic characterization of functions j with finite zeros such that the integral operator Sj is bounded on F2(C)using Hadamard’s factorization theorem.As an application,we obtain a complete characterization for such symbol functions j such that the Berezin transform of Sj is bounded while the operator Sj is not.Also,the corresponding problem in higher dimensions is considered.展开更多
The aim of this work is to study the Berezin quantization of a Gaussian state. The result is another Gaussian state that depends on a quantum parameter α, which describes the relationship between the classical and qu...The aim of this work is to study the Berezin quantization of a Gaussian state. The result is another Gaussian state that depends on a quantum parameter α, which describes the relationship between the classical and quantum vision. The compression parameter λ>0 is associated to the harmonic oscillator semigroup.展开更多
Let Ω be the unit ball or the polydisk of Cnand L2a(Ω) the Bergman space. In this paper we prove that if S is a finite sum of finite products of Toeplitz operators on L2a( Ω), then S is compact if and only if the B...Let Ω be the unit ball or the polydisk of Cnand L2a(Ω) the Bergman space. In this paper we prove that if S is a finite sum of finite products of Toeplitz operators on L2a( Ω), then S is compact if and only if the Berezin transform S(z) of S tends to zero as z→Ω.展开更多
基金Research partially supported by NNSF of China(11720101003)NSF of Guangdong Province(2018A030313512)+1 种基金Key projects of fundamental research in universities of Guangdong Province(2018KZDXM034)STU Scientific Research Foundation(NTF17009).
文摘We give a survey on the Berezin transform and its applications in operator theory. The focus is on the Bergman space of the unit disk and the Fock space of the complex plane. The Berezin transform is most effective and most successful in the study of Hankel and Toepltiz operators.
文摘In this paper, we analyze a class of bounded radial operators on the weighted Bergman space A2α(Bn, d Vα), we get that these kinds of operators are diagonal with respect to the standard orthonomal basis. We also investigate the connection between compactness of operators and the boundary behaviour of the corresponding Berezin transform. We further study a special class of radial operators, i.e., Toeplitz operators with a radial L1 symbol.
基金supported by the Chongqing Natural Science Foundation of China(No.cstc 2013jj B0050)
文摘Abstract This paper deals with the relationship between the positivity of the Fock Toeplitz operators and their Berezin transforms. The author considers the special case of the bounded radial function φ（z）=α ＋ be-α｜z｜2 ＋ ce-β｜z｜2, where a, b, c are real numbers and α,β are positive numbers. For this type of φ, one can choose these parameters such that the Berezin transform of is a nonnegative function on the complex plane, but the corresponding Toeplitz operator Tφ is not positive on the Fock space.
基金supported in part by the Natural Science Foundation of Tianjin City of China(Grant No.19JCQNJC14700).
文摘There is a singular integral operators Sj on the Fock space F2(C),which originated from the unitarily equivalent version of the Hilbert transform on L2(R).In this paper,we give an analytic characterization of functions j with finite zeros such that the integral operator Sj is bounded on F2(C)using Hadamard’s factorization theorem.As an application,we obtain a complete characterization for such symbol functions j such that the Berezin transform of Sj is bounded while the operator Sj is not.Also,the corresponding problem in higher dimensions is considered.
文摘The aim of this work is to study the Berezin quantization of a Gaussian state. The result is another Gaussian state that depends on a quantum parameter α, which describes the relationship between the classical and quantum vision. The compression parameter λ>0 is associated to the harmonic oscillator semigroup.
文摘Let Ω be the unit ball or the polydisk of Cnand L2a(Ω) the Bergman space. In this paper we prove that if S is a finite sum of finite products of Toeplitz operators on L2a( Ω), then S is compact if and only if the Berezin transform S(z) of S tends to zero as z→Ω.
