Berezin transform and Toeplitz operators on harmonic Bergman spaces

Abstract

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.

Boundedness of Berezin Transform on Herz Spaces

“…Theorem 5.2 expresses the growth estimates of the Berezin transforms on such Herz spaces. Similar results on the unit ball of R n can also be found in [1].…”

“…Motivated by these ideas of [1], we give a more complete result of (1.3) and the growth estimates of the Berezin transforms in the weighted pluriharmonic Bergman space case on the unit ball of C n . Now, we are about to state our main results as follows.…”

The Atomic Decomposition of Harmonic Bergman Functions, Dualities and Toeplitz Operators

“…We first have the following growth estimate of ||R x || 1 . By Lemma 3.1 of [2], we see that there exists a positive constant C, depending only on n, such that…”

“…Since each point evaluation is a bounded linear functional on b 2 for each x ∈ B, there exists a unique function R(x, ·) ∈ b 2 which has the following reproducing property: (1) f (x) = B f (y)R(x, y) dV (y) for all f ∈ b 2 . The kernel R(x, y) is the well-known harmonic Bergman kernel whose its explicit formula is given by R(x, y) = (n − 4)|x| 4 |y| 4 + (8x · y − 2n − 4)|x| 2 |y| 2 + n nV (B)(1 − 2x · y + |x| 2 |y| 2 ) 1+ n 2 , y ∈ B, where x · y denotes the usual inner product for points x, y ∈ R n .…”

“…The result [5,Theorem 3] of Coifman and Rochberg implies that every function in b 1 admits the following atomic decomposition: Given an integer m > 0, there exist a sequence {a j } in B and a constant C such that every f ∈ b 1 can be represented as (2) f (x) =…”