$m$-Berezin transform and compact operators

Abstract

ABSTRACT. m-Berezin transforms are introduced for bounded operators on the Bergman space of the unit ball. The norm of the m-Berezin transform as a linear operator from the space of bounded operators to L ∞ is found. We show that the m-Berezin transforms are commuting with each other and Lipschitz with respect to the pseudo-hyperbolic distance on the unit ball. Using the mBerezin transforms we show that a radial operator in the Toeplitz algebra is compact iff its Berezin transform vanishes on the boundary of the unit ball.

Covariant derivatives of the Berezin transform

“…The Lipschitz estimate (3) means, in particular, that X is uniformly continuous with respect to the Bergman metric; this was applied for Ω = D, the unit disc, by Suárez [28], and for Ω the unit ball of C n , n > 1, by Nam, Zheng and Zhong [25], in the study of Toeplitz algebras.…”

m-Berezin Transform on the Polydisk

“…Recently, they have been established for operators S on the Bergman space of the unit ball in [12]. We will show that for each m, B m S(z) is Lipschitz with respect to the pseudo-hyperbolic distance ρ(z, w).…”

C*-Algebra Generated by Angular Toeplitz Operators on the Weighted Bergman Spaces Over the Upper Half-Plane

“…The results of Suárez have been complemented and generalized to the weighted Bergman space on the unit ball in [1,2,10,15]. The above ∞ -closure of d 1 was characterized in [10].…”