Localization and compactness in Bergman and Fock spaces

Abstract

In this paper we study the compactness of operators on the Bergman space of the unit ball and on very generally weighted Bargmann-Fock spaces in terms of the behavior of their Berezin transforms and the norms of the operators acting on reproducing kernels. In particular, in the Bergman space setting we show how a vanishing Berezin transform combined with certain (integral) growth conditions on an operator T are sufficient to imply that the operator is compact. In the weighted Bargmann-Fock space setting we show that the reproducing kernel thesis for compactness holds for operators satisfying similar growth conditions. The main results extend the results of Xia and Zheng to the case of the Bergman space when 1 < p < ∞, and in the weighted Bargmann-Fock space setting, our results provide new, more general conditions that imply the work of Xia and Zheng via a more familiar approach that can also handle the 1 < p < ∞ case.2000 Mathematics Subject Classification. 32A36, 32A, 47B05, 47B35.

Localization and the Toeplitz algebra on the Bergman space

“…Indeed the approach in [6] explains why such results should hold true. The results in [6,13] certainly inspire further examinations of the inclusion relation…”

“…Recently, this idea was further explored in [6]. More specifically, in [6] Isralowitz, Mitkovski and Wick introduced the notion of weakly localized operators on the Bergman space. Below we give a slightly more refined version of their definition.…”

“…In [6], Isralowitz, Mitkovski and Wick showed that for A ∈ C * (A s ), condition (1.2) also implies that A is compact. Moreover, the introduction of the notion of weakly localized operators in [6] has the added virtue that it significantly simplifies the work necessary to obtain the above result. Indeed the approach in [6] explains why such results should hold true.…”

“…Moreover, the introduction of the notion of weakly localized operators in [6] has the added virtue that it significantly simplifies the work necessary to obtain the above result. Indeed the approach in [6] explains why such results should hold true. The results in [6,13] certainly inspire further examinations of the inclusion relation…”

Geometric Arveson-Douglas conjecture and holomorphic extension

“…Second, we use the idea that Toeplitz operators are "localized" (cf. [18][27]) and apply it on the variety, instead of the whole unit ball. Finally, we observe that the fact that the size of the hyperbolic balls tend to 0 uniformly as the centers tend to the boundary forces the Bergman reproducing kernels K z (w) to act "almost" like reproducing kernels on the quotient space.…”