arXiv subjects
Publications and source records attributed to Kehe Zhu.
Let $f$ and $g$ be functions, not identically zero, in the Fock space $F^2$ of $C_n$. We show that the product $T_fT_{\bar g}$ of Toeplitz operators on $F^2$ is bounded if and only if $f(z)=e^{q(z)}$ and $g(z)=ce^{-q(z)}$, where $c$ is a nonzero constant and $q$ is a linear polynomial.
We obtain sufficient conditions for a densely defined operator on the Fock space to be bounded or compact. Under the boundedness condition we then characterize the compactness of the operator in terms of its Berezin transform.
A sequence $Z$ in the complex plane $\C$ is called a zero sequence for the Fock space $F^p_α$ if there exists a function $f\in F^p_α$, not identically zero, such that $Z$ is the zero set of $f$, counting multiplicities. We show that there exist zero sequences $Z$ for $F^p_α$ with the following properties: (1) For any $a\in\C$ the sequence $Z\cup\{a\}$ is no longer a zero sequence for $F^p_α$; (2) the space $I_Z$ consisting of all functions in $F^p_α$ that vanish on $Z$ is one dimensional. These $Z$ are naturally called maximal zero sequences for $F^p_α$.
We show that there is only one Hilbert space of entire functions that is invariant under the action of naturally defined weighted translations.
We show that the $L^2$ integral mean on $r\D$ of an analytic function in the unit disk $\D$ with respect to the weighted area measure $(1-|z|^2)^α\,dA(z)$, where $-3\leα\le0$, is a logarithmically convex function of $r$ on $(0,1)$. We also show that the range $[-3,0]$ for $α$ is best possible.
It is well known that functions in the analytic Besov space $B_1$ on the unit disk $\D$ admits an integral representation $$f(z)=\ind\frac{z-w}{1-z\bar w}\,dμ(w),$$ where $μ$ is a complex Borel measure with $|μ|(\D)<\infty$. We generalize this result to all Besov spaces $B_p$ with $0 1$. We also obtain a version for Bergman and Fock spaces.
We obtain several new characterizations for the standard weighted Bergman spaces $A^p_α$ on the unit ball of $\cn$ in terms of the radial derivative, the holomorphic gradient, and the invariant gradient.
We study the $L^p$ boundedness and find the norm of a class of integral operators induced by the reproducing kernel of Fock spaces over $C^n$.
There has been a great deal of work done in recent years on weighted Bergman spaces $\apa$ on the unit ball $\bn$ of $\cn$, where $0 -1$. We extend this study in a very natural way to the case where $α$ is {\em any} real number and $0<p\le\infty$. This unified treatment covers all classical Bergman spaces, Besov spaces, Lipschitz spaces, the Bloch space, the Hardy space $H^2$, and the so-called Arveson space. Some of our results about integral representations, complex interpolation, coefficient multipliers, and Carleson measures are new even for the ordinary (unweighted) Bergman spaces of the unit disk.
We obtain new characterizations for Bergman spaces with standard weights in terms of Lipschitz type conditions in the Euclidean, hyperbolic, and pseudo-hyperbolic metrics. As a consequence, we prove optimal embedding theorems when an analytic function on the unit disk is symmetrically lifted to the bidisk.