THE BEREZIN TRANSFORM AND ITS APPLICATIONS

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.

单位球加权Bergman空间上的Berezin变换与径向算子（英文）

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.

Compact Operators on the Bergman Spaces of the Unit Ball and the Polydisk in C^n

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.

THE PRECISE NORM OF A CLASS OF FORELLI-RUDIN TYPE OPERATORS ON THE SIEGEL UPPER HALF SPACE

The precise L^(p) norm of a class of Forelli-Rudin type operators on the Siegel upper half space is given in this paper.The main result not only implies the upper L^(p) norm estimate of the Bergman projection,but also implies the precise L^(p) norm of the Berezin transform.

Commuting Toeplitz Operators with Harmonic Symbols on A^2(■,dμ)

For any given symmetric measure μ on the closed unit disk D, we apply the Berezin transform to characterizing semi-commuting and commuting Toeplitz operators with bounded harmonic symbols on A2（D, dμ）.

Boundary behavior of Gleason's problem in hyperbolic harmonic Bergman spaces

Necessary and sufficient conditions are obtained for the boundedness of Berezin transformation on Lebesgue space Lp(B,dVβ) in the real unit ball B in Rn. As an application, we prove that Gleason type problem is solvable in hyperbolic harmonic Bergman spaces. Furthermore we investigate the boundary behavior of the solutions of Gleason type problem.

Sarason’s Toeplitz Product Problem on the Fock–Sobolev Space

The Sarason＇s Toeplitz product problem asks when the Toeplitz product operator TuTv, with analytic symbols u and v, is bounded on Hilbert space of analytic functions. In this paper, we deal with this problem on the Fock-Sobolev space and have a complete solution that u = eq, v = Ce-q, where q is a linear complex polynomial and C is a nonzero constant.

Toeplitz Operators with BMO Symbols on the Weighted Bergman Space of the Unit Ball

In this paper we prove that the boundedness and compactness of Toeplitz operator with a BMOα1 symbol on the weighted Bergman space Aα2（Bn） of the unit ball is completely determined by the behavior of its Berezin transform, where ≥ -1 and n 〉 1.

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.

Commuting Toeplitz Operators on the Hardy Space of the Polydisk

In this paper, we show that two Toeplitz operators Tf and Tg on the Hardy space of the polydisk can commute if and only if the Berezin transform of the commutator [Tf, Tg] is n-harmonic.

Positive Toeplitz Operators on the Bergman Spaces of the Siegel Upper Half-Space

We characterize bounded and compact positive Toeplitz operators defined on the Bergman spaces over the Siegel upper half-space.

Toeplitz Operators with BMO Symbols of Several Complex Variables

In this note we prove that the boundedness and compactness of the Toeplitz operator on the Bergman space L2a （Bn） for several complex variables with a BMO1 symbol is completely determined by the boundary behavior of its Berezin transform.