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Guangfu Cao

Publications and source records attributed to Guangfu Cao.

10 recordsLinked to original sources
arXiv2025

The essential norm of Toeplitz operators between Bergman spaces induced by doubling weights

This paper investigates the essential norm of Toeplitz operators $\mathcal{T}_μ$ acting from the Bergman space $A_ω^p$ to $A_ω^q$ ($1 < p \leq q < \infty$) on the unit ball, where $μ$ is a positive Borel measure and $ω\in \mathcal{D}$ (a class of doubling weights). Leveraging the geometric properties of Carleson blocks and the structure of radial doubling weights, we establish sharp estimates for the essential norm in terms of the asymptotic behavior of $μ$ near the boundary. As a consequence, we resolve the boundedness-to-compactness transition for these operators when $1 < q < p<\infty$, showing that the essential norm vanishes exactly. These results generalize classical theorems for the unweighted Bergman space ($ω\equiv 1$) and provide a unified framework for studying Toeplitz operators under both radial and non-radial doubling weights in higher-dimensional settings.

math.FA↗
arXiv2023

Composition operators with closed range on the Dirichlet space

It is well known that the composition operator on Hardy or Bergman space has a closed range if and only if its Navanlinna counting function induces a reverse Carleson measure. Similar conclusion is not available on the Dirichlet space. Specifically, the reverse Carleson measure is not enough to ensure that the range of the corresponding composition operator is closed. However, under certain assumptions, we in this paper set the necessary and sufficient condition for a composition operator on the Dirichlet space to have closed range.

math.FA↗
arXiv2023

Evaluation functions and composition operators on Banach spaces of holomorphic functions

Let $B(Ω)$ be the Banach space of holomorphic functions on a bounded connected domain $Ω$ in $\mathbb C^n$, which contains the ring of polynomials on $Ω$. In this paper, we first establish a criterion for $B(Ω)$ to be reflexive via evaluation functions on $B(Ω)$, that is, $B(Ω)$ is reflexive if and only if the evaluation functions span the dual spaces $(B(Ω))^{*} $. Moreover, under suitable assumptions on $Ω$ and $B(Ω)$, we establish a characterization of the composition operator $C_φ$ to be a Fredholm operator on $B(Ω)$ via the property of the holomorphic self-map $φ:Ω\toΩ$. Our new approach utilizes the symbols of composition operators to construct a linearly independent function sequence, which bypasses the use of boundary behavior of reproducing kernels as those may not be applicable in our general setting.

math.CV↗
arXiv2022

Composition operators on Hardy-Sobolev spaces with bounded reproducing kernels

For any real $β$ let $H^2_β$ be the Hardy-Sobolev space on the unit disc $\mathbb{D}$. $H^2_β$ is a reproducing kernel Hilbert space and its reproducing kernel is bounded when $β>1/2$. In this paper, we characterize that for a non-constant analytic function $φ:\mathbb{D}\to\mathbb{D}$, when the composition operator $C_{φ}$ on $H^{2}_{β}$ is Fredholm. For $1/2<β<1$, we also prove that $C_{φ}$ has dense range in $H_{β}^{2}$ if and only if the polynomials are dense in a certain Dirichlet space of the domain $φ(\mathbb{D})$. It follows that if the range of $C_{φ}$ is dense in $H_{β}^{2}$, then $φ$ is a weak-star generator of $H^{\infty}$, although the conclusion is false for the classical Dirichlet space $\mathfrak{D}$. Moreover, we study the relation between the density of the rang of $C_{φ}$ and the cyclic vector of the multiplier $M_φ^β.$

math.CV↗
arXiv2022

Composition Operators on Dirichlet Spaces over the Half-plane

As continuation of the study of polynomial approximation and composition operators on Dirichlet spaces of unit disk, which has settled a problem posed by Cima in 1976, the present paper aims to consider the case of the unbounded domains, such as the half-plane. Specifically, we may obtain the rational approximations in the Dirichlet spaces and characterize the composition operators which has dense range on the Dirichlet spaces over the half-plane. Moreover, this paper also considers the relationship between the Dirichlet spaces and Hardy spaces on half-plane.

math.CV↗
arXiv2021

Products of Toeplitz and Hankel Operators on Fock-Sobolev Spaces

In this paper, we investigate the boundedness of Toeplitz product $T_{f}T_{g}$ and Hankel product $H_{f}^{*} H_{g}$ on Fock-Sobolev space for two polynomials $f$ and $g$ in $z,\overline{z}\in\mathbb{C}^{n}$. As a result, the boundedness of Toeplitz operator $T_{f}$ and Hankel operator $H_{f}$ with the polynomial symbol $f$ in $z,\overline{z}\in\mathbb{C}^{n}$ is characterized.

math.FA↗
arXiv2021

Boundedness criterion for integral operators on the fractional Fock-Sobolev spaces

We provide a boundedness criterion for the integral operator $S_φ$ on the fractional Fock-Sobolev space $F^{s,2}(\mathbb C^n)$, $s\geq 0$, where $S_φ$ (introduced by Kehe Zhu) is given by \begin{eqnarray*} S_φF(z):= \int_{\mathbb{C}^n} F(w) e^{z \cdot\bar{w}} φ(z- \bar{w}) dλ(w) \end{eqnarray*} with $φ$ in the Fock space $F^2(\mathbb C^n)$ and $dλ(w): = π^{-n} e^{-|w|^2} dw$ the Gaussian measure on the complex space $\mathbb{C}^{n}$. This extends the recent result in Cao--Li--Shen--Wick--Yan. The main approach is to develop multipliers on the fractional Hermite-Sobolev space $W_H^{s,2}(\mathbb R^n)$.

math.CV↗
arXiv2020

A Boundedness Criterion for Singular Integral Operators of convolution type on the Fock Space

We show that for an entire function $φ$ belonging to the Fock space ${\mathscr F}^2(\mathbb{C}^n)$ on the complex Euclidean space $\mathbb{C}^n$, the integral operator \begin{eqnarray*} S_φF(z)=\int_{\mathbb{C}^n} F(w) e^{z \cdot\bar{w}} φ(z- \bar{w})\,dλ(w), \ \ \ \ \ z\in \mathbb{C}^n, \end{eqnarray*} is bounded on ${\mathscr F}^2(\mathbb{C}^n)$ if and only if there exists a function $m\in L^{\infty}(\mathbb{R}^n)$ such that $$ φ(z)=\int_{\mathbb{R}^n} m(x)e^{-2\left(x-\frac{i}{2} z \right)\cdot \left(x-\frac{i}{2} z \right)} dx, \ \ \ \ \ \ z\in \mathbb{C}^n. $$ Here $dλ(w)= π^{-n}e^{-\left\vert w\right\vert^2}dw$ is the Gaussian measure on $\mathbb C^n$. With this characterization we are able to obtain some fundamental results including the normaility, the algebraic property, spectrum and compactness of this operator $S_φ$. Moreover, we obtain the reducing subspaces of $S_φ$. In particular, in the case $n=1$, we give a complete solution to an open problem proposed by K. Zhu for the Fock space ${\mathscr F}^2(\mathbb{C})$ on the complex plane ${\mathbb C}$ (Integr. Equ. Oper. Theory {\bf 81} (2015), 451--454).

math.CV↗
arXiv2017

Spectral theory of multiplication operators on Hardy-Sobolev spaces

For a pointwise multiplier $φ$ of the Hardy-Sobolev space $H^2_β$ on the open unit ball $\bn$ in $\cn$, we study spectral properties of the multiplication operator $M_φ: H^2_β\to H^2_β$. In particular, we compute the spectrum and essential spectrum of $M_φ$ and develop the Fredholm theory for these operators.

math.FA↗