Bounded Toeplitz Products on the Bergman Space of the Unit Ball in $$\mathbb{C}^n $$

Abstract

We investigate necessary and sufficient conditions for boundedness of the operator T f Tg on the Bergman space of the unit ball Bn for n ≥ 1, where T f is the Toeplitz operator. Those conditions are related to boundedness of the Berezin transform of symbols f and g. We construct the inner product formula which plays a crucial role in proving the sufficiency of the conditions. (2000). Primary 47B35. Mathematics Subject Classification

Bounded Toeplitz and Hankel Products on the Weighted Bergman Spaces of the Unit Ball

“…They also obtained analogous results for the Bergman space in the polydisk [13], for the weighted Bergman spaces in the unit disk [15] and for the weighted Bergman spaces in the unit ball [14]. Similar conditions for the weighted Bergman spaces in the unit ball were obtained by Park [7], while in [8] Pott and Strouse gave the related results for the space A 2 α in the unit disk. Recently, Miao [4] generalized the results of Stroethoff and Zheng to the weighted Bergman spaces A p α for all p > 1.…”

Toeplitz and Hankel products on Bergman spaces of the unit ball

“…Stroethoff and Zheng showed the analogous result on the Bergman space of the polydisk in [8] and on the weighted Bergman space of the unit disk in [9] and the unit ball in [10]. In [11], Park gave the analogous result for Toeplitz products on the Bergman space of the unit ball. In On the Bergman space, little is known concerning the products H * f H g or H f H * g for f, g ∈ L 2 (D, dA).…”

Bounded Toeplitz products on the weighted Bergman spaces of the unit ball

“…In another paper [3], they established the corresponding results for the Bergman space of the polydisk. Park also proved Stroethoff and Zheng's results for the Bergman space A 2 0 of the unit ball in [1].…”