Compact Operators via the Berezin Transform

In this paper we prove that if S equals a finite sum of finite products of Toeplitz operators on the Bergman space of the unit disk, then S is compact if and only if the Berezin transform of S equals 0 on the boundary of the disk. This result is new even when S equals a single Toeplitz operator. Our main result can be used to prove, via a unified approach, several previously known results about compact Toeplitz operators, compact Hankel operators, and appropriate products of these operators.Comment: 15 pages. To appear in Indiana University Mathematics Journal. For more information, see http://math.sfsu.edu/axler/CompactBerezin.htm

Bounded Toeplitz products on Bergman spaces of the unit ball

AbstractWe consider the question for which square integrable analytic functions f and g on the unit ball the densely defined products TfTg¯ are bounded on the weighted Bergman spaces. We prove results analogous to those we obtained in the setting of the unit disk and the polydisk

Similarity of analytic Toeplitz operators on the Bergman spaces

AbstractIn this paper we give a function theoretic similarity classification for Toeplitz operators on weighted Bergman spaces with symbol analytic on the closure of the unit disk

Commuting Toeplitz operators on the bidisk

AbstractA necessary and sufficient condition is obtained for two Toeplitz operators to be commuting on the Hardy space of the bidisk. The main tool is the Berezin transform and the harmonic extension

The spectral picture of Bergman Toeplitz operators with harmonic polynomial symbols

In this paper, it is shown that some new phenomenon related to the spectra of Toeplitz operators with bounded harmonic symbols on the Bergman space. On the one hand, we prove that the spectrum of the Toeplitz operator with symbol ${\bar{z}+p}$ is always connected for every polynomial $p$ with degree less than $3$. On the other hand, we show that for each integer $k$ greater than $2$, there exists a polynomial $p$ of degree $k$ such that the spectrum of the Toeplitz operator with symbol ${\bar{z}+p}$ has at least one isolated point but has at most finitely many isolated points. Then these results are applied to obtain a new class of non-hyponormal Toeplitz operators with bounded harmonic symbols on the Bergman space for which Weyl's theorem holds.Comment: 21 page

Helton-Howe Trace, the Connes-Chern character and Quantization

We study the Helton-Howe trace and the Connes-Chern character for Toeplitz operators on weighted Bergman spaces via the idea of quantization. We prove a local formula for the large $t$-limit of the Connes-Chern character as the weight goes to infinity. And we show that the Helton-Howe trace of Toeplitz operators is independent of the weight $t$ and obtain a local formula for the Helton-Howe trace for all weighted Bergman spaces using harmonic analysis and quantization.Comment: 92 page