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Zhangjian Hu

Publications and source records attributed to Zhangjian Hu.

11 recordsLinked to original sources
arXiv2026

Absolutely summing Hankel operators on Fock spaces and the Berger-Coburn phenomenon

In this paper, for $1 \leq p, r < \infty$ we characterize those symbols $f$ so that the induced Hankel operators $H_f$ are $r$-summing from Fock spaces $F^p_α$ to $L^p_α$. The main result shows that the $r$-summing norm of $H_f$ is equivalent to the $\mathrm{IDA}^{κ, p}$-norm of $f$, where $κ$ is a positive number determined by $p$ and $r$, and the $\mathrm{IDA}$ space is as in [13]. As some application, we discuss the Berger-Coburn phenomenon for $r$-summing Hankel operators on Fock spaces.

math.FA↗
arXiv2025

Absolutely Summing Toeplitz operators on Bergman spaces in the unit ball of $\mathbb{C}^n$

In this paper, for $p> 1 $ and $r \ge 1$ we provide a complete characterization of the positive Borel measures $μ$ on the unit ball $\B_n$ of $\mathbb {C}^n$ for which the induced Toeplitz operator $T_μ$ is $r$-summing on the Bergman space $A^{p}$. We prove that the $r$-summing norm of $T_μ: A^p\to A^p$ is equivalent to $\|\widetildeμ\|_{L^κ(dλ)}$, where $κ$ is a positive number determined by $p$ and $r$. As some preliminary, we describe when a Carleson embedding $J_μ: A^p \to L^q(μ) (1\le p, q\le 2)$ is $r$-summing, which extends the main result in [B. He, et al, Absolutely summing Carleson embeddings on Bergman spaces, Adv. Math., 439, 109495 (2024)].

math.FA↗
arXiv2025

Absolutely Summing Toeplitz operators on Fock spaces

For $1\le p<\infty$, let $F^p_φ$ be the Fock spaces on ${\mathbb C}^n$ with the weight function $φ$ that \(φ\in {\mathcal{C}}^{2}\left( {\mathbb{C}}^{n}\right)\) is real-valued and satisfies $ m{ω}_{0} \leq d{d}^{c}φ\leq M{ω}_{0} $ for two positive constants \(m\) and \(M\), \({ω}_{0} = d{d}^{c}{\left| z\right| }^{2}\) is the Euclidean Kähler form on \({\mathbb{C}}^{n}\), \({d}^{c} = \frac{\sqrt{-1}}{4}\left( {\bar{\partial } - \partial }\right)\). In this paper, we completely characterize those positive Borel measure $μ$ on ${\mathbb C}^n$ so that the induced Toeplitz operators $T_μ$ is $r$-summing on $F_φ^{p}$ for $r \ge 1$.

math.FA↗
arXiv2023

Bounded, compact and Schatten class Hankel operators on Fock-type spaces

In this paper, we consider Hankel operators, with locally integrable symbols, densely defined on a family of Fock-type spaces whose weights are $C^3$-logarithmic growth functions with mild smoothness conditions. It is shown that a Hankel operator is bounded on such a Fock space if and only if its symbol function has bounded distance to analytic functions BDA which is initiated by Luecking(J. Funct. Anal. 110:247-271, 1992). We also characterize the compactness and Schatten class membership of Hankel operators. Besides, we give characterizations of the Schatten class membership of Toeplitz operators with positive measure symbols for the small exponent $0<p<1$. Our proofs depend strongly on the technique of Hömander's $L^2$ estimates for the $\overline{\partial}$ operator and the decomposition theory of BDA spaces as well as integral estimates involving the reproducing kernel.

math.FA↗
arXiv2023

On the Berger-Coburn phenomenon

In their previous work, the authors proved the Berger-Coburn phenomenon for compact and Schatten $S_p$ class Hankel operators $H_f$ on generalized Fock spaces when $1<p<\infty$, that is, for a bounded symbol $f$, if $H_f$ is a compact or Schatten class operator, then so is $H_{\bar f}$. More recently J.~Xia has provided a simple example that shows that there is no Berger-Coburn phenomenon for trace class Hankel operators on the classical Fock space $F^2$. Using Xia's example, we show that there is no Berger-Coburn phenomena for Schatten $S_p$ class Hankel operators on generalized Fock spaces $F^2_φ$ for any $0<p\le 1$. Our approach is based on the characterization of Schatten class Hankel operators while Xia's approach is elementary and heavily uses the explicit basis vectors of $F^2$, which cannot be found for the weighted Fock spaces that we consider. We also formulate four open problems.

math.FA↗
arXiv2022

IDA and Hankel operators on Fock spaces

We introduce a new space IDA of locally integrable functions whose integral distance to holomorphic functions is finite, and use it to completely characterize boundedness and compactness of Hankel operators on weighted Fock spaces. As an application, for bounded symbols, we show that the Hankel operator $H_f$ is compact if and only if $H_{\bar f}$ is compact, which complements the classical compactness result of Berger and Coburn. Motivated by recent work of Bauer, Coburn, and Hagger, we also apply our results to the Berezin-Toeplitz quantization.

math.FA↗
arXiv2021

Fredholm Toeplitz operators on doubling Fock spaces

Recently the authors characterized the Fredholmn properties of Toeplitz operators on weighted Fock spaces when the Laplacian of the weight function is bounded below and above. In the present work the authors extend their characterization to doubling Fock spaces with a subharmonic weight whose Laplacian is a doubling measure. The geometry induced by the Bergman metric for doubling Fock spaces is much more complicated than that of the Euclidean metric used in all the previous cases to study Fredholmness, which leads to considerably more involved calculations.

math.FA↗
arXiv2021

Toeplitz operators on Bergman spaces with exponential weights

In this paper, we focus on the weighted Bergman spaces $A_φ^{p}$ in $\mathbb{D}$ with $φ\in\mathcal{W}_{0}$. We first give characterizations of those finite positive Borel measures $μ$ in $\mathbb{D}$ such that the embedding $A_φ^{p}\subset L_μ^{q}$ is bounded or compact for $0<p,q<\infty$. Then we describe bounded or compact Toeplitz operators $T_μ$ from one Bergman space $A_φ^{p}$ to another $A_φ^{q}$ for all possible $0<p,q<\infty$. Finally, we characterize Schatten class Toeplitz operators on $A_φ^{2}$.

math.FA↗
arXiv2017

Localization and compactness of Operators on Fock Spaces

For $0<p\leq\infty$, let $F^{p}_φ$ be the Fock space induced by a weight function $φ$ satisfying $ dd^c φ\simeq ω_0$. In this paper, given $p\in (0, 1]$ we introduce the concept of weakly localized operators on $ F^{p}_φ$, we characterize the compact operators in the algebra generated by weakly localized operators. As an application, for $0<p<\infty$ we prove that an operator $T$ in the algebra generated by bounded Toeplitz operators with $\textrm{BMO}$ symbols is compact on $F^p_φ$ if and only if its Berezin transform satisfies certain vanishing property at $\infty$. In the classical Fock space, we extend the Axler-Zheng condition on linear operators $T$, which ensures $T$ is compact on $F^p_α$ for all possible $0<p<\infty$.

math.CV↗