Berezin transform

Abstract: In this paper we prove that if S equals a finite sum of finite productsof Toeplitz operators on the Bergman space of the unit disk, thenS is compact if and only if the Berezin transform of S equals 0 on∂D. This result is new even when S equals a single Toeplitz operator.Our main result can be used to prove, via a unified approach, severalpreviously known results about compact Toeplitz operators, compactHankel operators, and appropriate products of these operators. 1 Introduction Let dA denote Lebesgue area measure on the unit disk D, normalizedso that the measure of D equals 1. The Bergman space L 2a is the Hilbertspace consisting of the analytic functions on D that are also in L 2 (D,dA).For z ∈ D, the Bergman reproducing kernel is the function K z ∈ L 2a suchthatf(z) = hf,K z ifor every f ∈ L 2a . The normalized Bergman reproducing kernel k z is thefunction K z /kK z k 2 . Here, as elsewhere in this paper, the norm k k 2 and theinner product h , i are taken in the space L 2 (D,dA).For S a bounded operator on L

Abstract: In this paper we prove that if S equals a finite sum of finite products of Toeplitz operators on the Bergman space of the unit disk, then S is compact if and only if the Berezin transform of S equals 0 on the boundary of the disk. This result is new even when S equals a single Toeplitz operator. Our main result can be used to prove, via a unified approach, several previously known results about compact Toeplitz operators, compact Hankel operators, and appropriate products of these operators.

Abstract: We characterize bounded and compact weighted composition operators acting between weighted Bergman spaces and between Hardy spaces. Our results use certain integral transforms that generalize the Berezin transform. We also estimate the essential norms of these operators. As applications, we characterize bounded and compact pointwise multiplication operators between weighted Bergman spaces and estimate their essential norms.

Abstract: This article is a review on Berezin-Toeplitz operator and Berezin-Toeplitz deformation quantization for compact quantizable Kahler manifolds. The basic objects, concepts, and results are given. This concerns the correct semiclassical limit behaviour of the operator quantization, the unique Berezin-Toeplitz deformation quantization (star product), covariant and contravariant Berezin symbols, and Berezin transform. Other related objects and constructions are also discussed.