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Kehe Zhu

Publications and source records attributed to Kehe Zhu.

At least 19 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↗
arXiv2026

Deep zero problems and the HRT conjecture

We investigate a "deep zero problem" proposed by Hedenmalm. We show that there is a natural connection between Hedenmalm's problem and the classical HRT conjecture in time-frequency analysis. This connection allows us to show that Hedenmalm's problem 5.2 in [5] as well as some of its natural analogs have affirmative answers.

math.FA↗
arXiv2025

Canonical integral operators on the Fock space II

In \cite{DZ3} we introduced and studied a two-parameter family of integral operators $T^{(s,t)}$ on the Fock space $F^2$ of the complex plane. Under the inverse Bargmann transform, these operators include the classical {\it linear canonical transforms} in mathematical physics as special cases, so we called $T^{(s,t)}$ {\it canonical linear operators} on the Fock space. In this paper we continue the study of these operators. We show that when a canonical linear operator $T^{(s,t)}$ is compact, it actually belongs to the Schatten class $S_p$ for all $p>0$. In this case, we find all singular values, determine the $S_p$ norm, and obtain a trace formula for $T^{(s,t)}$. We also show that the boundedness (and a natural version of compactness) of $T^{(s,t)}$ on $F^p$ for any given $p\in(0,\infty]$ is equivalent to the boundedness (and compactness) of $T^{(s,t)}$ on $F^2$. Our analysis is based on estimates and computations with the integral kernel of $T^{(s,t)}$, which also yield some interesting results about the Berezin transform and the bivariate Berezin transform of $T^{(s,t)}$.

math.FA↗
arXiv2025

New characterizations for Fock spaces

We show that the maximal Fock space $F^\infty_\alpha$ on $C^n$ is a Lipschitz space, that is, there exists a distance $d_\alpha$ on $C^n$ such that an entire function $f$ on $C^n$ belongs to $F^\infty_\alpha$ if and only if $$|f(z)-f(w)|\le Cd_\alpha(z,w)$$ for some constant $C$ and all $z,w\in C^n$. This can be considered the Fock space version of the following classical result in complex analysis: a holomorphic function $f$ on the unit ball $B_n$ in $C^n$ belongs to the Bloch space if and only if there exists a positive constant $C$ such that $|f(z)-f(w)|\le C\beta(z,w)$ for all $z,w\in B_n$, where $\beta(z,w)$ is the distance on $B_n$ in the Bergman metric. We also present a new approach to Hardy-Littlewood type characterizations for $F^p_\alpha$.

math.CV↗
arXiv2025

Embedding and compact embedding between Bergman and Hardy spaces

For Hardy spaces and weighted Bergman spaces on the open unit ball in ${\mathbb C}^n$, we determine exactly when $A^p_\alpha\subset H^q$ or $H^p\subset A^q_\alpha$, where $0<q<\infty$, $0<p<\infty$, and $-\infty<\alpha<\infty$. For each such inclusion we also determine exactly when it is a compact embedding. Although some special cases were known before, we are able to completely cover all possible cases here. We also introduce a new notion called {\it tight fitting} and formulate a conjecture in terms of it, which places several prominent known results about contractive embeddings in the same framework.

math.CV↗
arXiv2023

Sub-Bergman Hilbert spaces on the unit disk III

For a bounded analytic function $\varphi$ on the unit disk $\D$ with $\|\varphi\|_\infty\le1$ we consider the defect operators $D_\varphi$ and $D_{\overline\varphi}$ of the Toeplitz operators $T_\varphi$ and $T_{\overline\varphi}$, respectively, on the weighted Bergman space $A^2_\alpha$. The ranges of $D_\varphi$ and $D_{\overline\varphi}$, written as $H(\varphi)$ and $H(\overline\varphi)$ and equipped with appropriate inner products, are called sub-Bergman spaces. We prove the following three results in the paper: for $-1<\alpha\le0$ the space $H(\varphi)$ has a complete Nevanlinna-Pick kernel if and only if $\varphi$ is a M\"{o}bius map; for $\alpha>-1$ we have $H(\varphi)=H(\overline\varphi)=A^2_{\alpha-1}$ if and only if the defect operators $D_\varphi$ and $D_{\overline\varphi}$ are compact; and for $\alpha>-1$ we have $D^2_\varphi(A^2_\alpha)= D^2_{\overline\varphi}(A^2_\alpha)=A^2_{\alpha-2}$ if and only if $\varphi$ is a finite Blaschke product. In some sense our restrictions on $\alpha$ here are best possible.

math.CV↗
arXiv2022

Canonical integral operators on the Fock space

In this paper we introduce and study a two-parameter family of integral operators on the Fock space $F^2(C)$. We determine exactly when these operators are bounded and when they are unitary. We show that, under the Bargmann transform, these operators include the classical linear canonical transforms as special cases. As an application, we obtain a new unitary projective representation for the special linear group $SL(2,R)$ on the Fock space.

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arXiv2020

Sarason Toeplitz product problem for a class of Fock spaces

Sarason Toeplitz product problem asks when the operator TuTv is bounded on various Hilbert spaces of analytic functions, where u and v are analytic. The problem is highly nontrivial for Toeplitz operators on the Hardy space and the Bergman space (even in the case of the unit disk). In this paper, we provide a complete solution to the problem for a class of Fock spaces on the complex plane. In particular, this generalizes an earlier result of Cho, Park, and Zhu.

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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.

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arXiv2016

The Fourier and Hilbert transforms under the Bargmann transform

There is a canonical unitary transformation from $L^2(\R)$ onto the Fock space $F^2$, called the Bargmann transform. We study the action of the Bargmann transform on several classical integral operators on $L^2(\R)$, including the fractional Fourier transform, the fractional Hilbert transform, and the wavelet transform.

math.CV↗
arXiv2015

Towards a dictionary for the Bargmann transform

There is a canonical unitary transformation from $L^2(\R)$ onto the Fock space $F^2$, called the Bargmann transform. The purpose of this article is to translate some important results and operators from the context of $L^2(\R)$ to that of $F^2$. Examples include the Fourier transform, the Hilbert transform, Gabor frames, pseudo-differential operators, and the uncertainty principle.

math.FA↗
arXiv2015

Weighted BMO and Hankel operators between Bergman spaces

We introduce a family of weighted BMO and VMO spaces for the unit ball and use them to characterize bounded and compact Hankel operators between different Bergman spaces. In particular, we resolve two problems left open by S. Janson in 1988 and R. Wallsten in 1990.

math.CV↗
arXiv2013

Geometric spectral theory for compact operators

We introduce a notion of joint spectrum for a tuple of compact operators on a separable Hilbert space and show that in many situations these operators commute if and only if the joint spectrum consists of countably many, locally finite, complex hyperplanes. In particular, we show that normal matrices (of the same size) $A_1,\cdots,A_n$ commute if and only if the polynomial $\det(z_1A_1+\cdots+z_nA_n+I)$ is completely reducible, that is, it can be factored into a product of linear polynomials.

math.FA↗
arXiv2013

Circle packing and interpolation in Fock spaces

It was shown by James Tung in 2005 that if a sequence $Z=\{z_n\}$ of points in the complex plane satisfies $$\inf_{n\not=m}|z_n-z_m|>2/\sqrtα,$$ then $Z$ is a sequence of interpolation for the Fock space $F^p_α$. Using results from circle packing, we show that the constant above can be improved to $$\sqrt{2π/(\sqrt3\,α)},$$ which is strictly smaller than $2/\sqrtα$. A similar result will also be obtained for sampling sequences.

math.CV↗
arXiv2013

Frames and operators in Schatten classes

Let $T$ be a compact operator on a separable Hilbert space $H$. We show that, for $2\le p<\infty$, $T$ belongs to the Schatten class $S_p$ if and only if $\{\|Tf_n\|\}\in \ell^p$ for \emph{every} frame $\{f_n\}$ in $H$; and for $0 \}$ and the double-indexed sequence $\{ \}$.

math.FA↗
arXiv2012

Fock-Sobolev spaces and their Carleson measures

We consider the Fock-Sobolev space $F^{p,m}$ consisting of entire functions $f$ such that $f^{(m)}$, the $m$-th order derivative of $f$, is in the Fock space $F^p$. We show that an entire function $f$ is in $F^{p,m}$ if and only if the function $z^mf(z)$ is in $F^p$. We also characterize the Carleson measures for the spaces $F^{p,m}$, establish the boundedness of the weighted Fock projection on appropriate $L^p$ spaces, identify the Banach dual of $F^{p,m}$, and compute the complex interpolation space between two $F^{p,m}$ spaces.

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