Trace Formula of Semicommutators

Abstract

For weighted Bergman spaces on the unit disk, we give trace formulas of semicommutators of Toeplitz operators with ${\mathcal{C}}^2(\overline{\mathbb{D}})$ symbols. We generalize this formula to weighted Bergman spaces on the unit ball in higher dimensions. Applications and examples on the Hankel operators are also discussed.

40 pages. This is an article independent of Version 2 of arXiv:2004.04337. This article together with Version 2 of arXiv:2004.04337 supersedes Version 1 of arXiv:2004.04337. The results on trace of semicommutators on the unit disk together with related topics in Version 1 of arXiv:2004.04337, are included in this article and generalized to unit balls in ${\mathbb{C}}^n$