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Dechao Zheng

Publications and source records attributed to Dechao Zheng.

13 recordsLinked to original sources
arXiv2026

Localization operators on Bergman and Fock spaces

We introduce localization operators on weighted Bergman and Fock spaces and show that, under a natural scaling of symbols and window functions, localization operators on the weighted Bergman space $A_{\beta r^2}^2$ converge, in the weak sense, to localization operators on the Fock space $F_{\beta}^2$ as $r\to\infty$. From this we derive several applications, including one about sharp norm estimates for certain Toeplitz operators on Fock spaces, one about windowed Berezin transforms for weighted Bergman spaces, and another about Szeg\"{o}-type theorems for localization operators on weighted Bergman spaces.

math.FA↗
arXiv2024

Multiplication operators on the Bergman space of bounded domains in C^d

In this paper we study multiplication operators on Bergman spaces of high dimensional bounded domains and those von Neumann algebras induced by them via the geometry of domains and function theory of their symbols. In particular, using local inverses and $L^2_a$-removability, we show that for a holomorphic proper map $Φ=(ϕ_1, ϕ_2, \cdots , ϕ_d)$ on a bounded domain $Ω$ in $\mathbb{C}^{d}$, the dimension of the von Neumann algebra $\mathcal{V}^*(Φ,Ω) $ consisting of bounded operators on the Bergman space $L_a^2(Ω)$, which commute with both $ M_{ϕ_j}$ and its adjoint $M_{ϕ_j}^*$ for each $j$, equals the number of components of the complex manifold $\mathcal{S}_{Φ}= \{(z,w)\in Ω^2: Φ(z)=Φ(w),\, z\not\in Φ^{-1}(Φ(Z))\},$ where $Z$ is the zero variety of the Jacobian $JΦ$ of $ Φ.$ This extends the main result in \cite{DSZ} in high dimensional complex domains. Moreover we show that the von Neumann algebra $\mathcal{V}^*(Φ,Ω) $ may not be abelian in general although Douglas, Putinar and Wang \cite{DPW} showed that $\mathcal{V}^*(Φ,\mathbb{D})$ for the unit disk $\mathbb{D}$ is abelian.

math.OA↗
arXiv2022

Helton-Howe Trace, 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.

math.FA↗
arXiv2022

Trace Formula of Semicommutators

For weighted Bergman spaces on the unit disk, we give trace formulas of semicommutators of Toeplitz operators with $\mathscr{C}^2(\overline{\mathbb{D}})$ 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.

math.CV↗
arXiv2021

Toeplitz operators on the Fock space via the Fourier transform

In sprite by Berger-Coburn theorems and their conjecture in \cite{Coburn1994}, we use the Fourier transform to decompose $ T_{g}$ as an infinite sum of Toeplitz operators with symbols which have compact support in the frequency domain. As a consequence, we obtain a sufficient condition for $ T_{g}$ to be bounded in terms of the Carleson measure conditions defined by the heat transform of the symbol $g$. Moreover the decomposition of a Toeplitz operator leads us to get easily understanding that for a bounded function $g$, if its Berezin transform vanishes at infinity, then the Toeplitz operator $T_g$ is compact \cite{Eng} and the Toeplitz algebra generated by Toeplitz operators with symbols in $L^{\infty}$ is indeed generated by Toeplitz operators with symbols which on uniformly continuous on ${\mathbb C}^n$ \cite{Bauer2012}.Further, we will apply our decomposition theory for a Toeplitz operator to estimate the Schatten $p$-norm of the product of two Toeplitz operators.

math.FA↗
arXiv2020

Essentially commuting dual truncated Toeplitz operators

In this paper, we completely characterize when two dual truncated Toeplitz operators are essentially commuting and when the semicommutator of two dual truncated Toeplitz operators is compact. Our main idea is to study dual truncated Toeplitz operators via Hankel operators, Toeplitz operators and function algebras.

math.FA↗
arXiv2020

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.

math.FA↗
arXiv2019

Toeplitz algebra on the Fock space

The Fock space consists of all entire functions which are square integrable with respect to Gauss measure. The Toeplitz algebra is the C*-algebra generated by the Toeplitz operator with bounded symbol on the Fock space. In this paper, we study the Toeplitz algebra on the Fock space.

math.FA↗
arXiv2009

Multiplication operators on the Bergman space via analytic continuation

In this paper, using the group-like property of local inverses of a finite Blaschke product $ϕ$, we will show that the largest $C^*$-algebra in the commutant of the multiplication operator $M_ϕ$ by $ϕ$ on the Bergman space is finite dimensional, and its dimension equals the number of connected components of the Riemann surface of $ϕ^{-1}\circϕ$ over the unit disk. If the order of the Blaschke product $ϕ$ is less than or equal to eight, then every $C^*$-algebra contained in the commutant of $M_ϕ$ is abelian and hence the number of minimal reducing subspaces of $M_ϕ$ equals the number of connected components of the Riemann surface of $ϕ^{-1}\circϕ$ over the unit disk.

math.FA↗
arXiv1998

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.

math.FA↗
arXiv1996

Products of Block Toeplitz operators

In this paper we characterize when the product of two block Toeplitz operators is a compact perturbation of a block Toeplitz operator on the Hardy space of the open unit disk. Necessary and sufficient conditions are given for the commutator of two block Toeplitz operators to be compact.

math.FA↗
arXiv1996

The semi-commutator of Toeplitz operators on the bidisc

In this paper we characterize when the semi-commutator $T_fT_g-T_{fg}$ of two Toeplitz operators $T_f$ and $T_g$ on the Hardy space of the bidisc is zero. We also show that there is no nonzero finite rank semi-commutator on the bidisc. Furthermore explicit examples of compact semi-commutators with symbols continuous on the bitorus $T^2$ are given.

math.FA↗