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Zeljko Cuckovic

Publications and source records attributed to Zeljko Cuckovic.

At least 19 recordsLinked to original sources
arXiv2025

Fixed points of the Berezin transform on Fock-type spaces

We study the fixed points of the Berezin transform on the Fock-type spaces $F_m^2$ with the weight $e^{-|z|^m}, m > 0.$ It is known that the Berezin transform is well-defined on the polynomials in $z$ and $\overline{z}$. In this paper we focus on the polynomial fixed points and we show that these polynomials must be harmonic, except possibly for countably many $m \in (0, \infty).$ We also show that, in some particular cases, the fixed point polynomials are harmonic for all $m.$

math.CV↗
arXiv2025

Pointwise lower bounds in growth spaces with little o conditions

Pointwise lower bounds on the open unit disc $\bbD$ for the sum of the moduli of two analytic functions $f$ and $g$ (or their derivatives) are known in several cases, like $f,g$ belonging to the Bloch space $\cB$, $BMOA$ or the weighted Hardy space $H_\omega^\infty$. We find complementary results of Ramey-Ullrich and Abakumov-Doubtsov for functions with little o conditions.

math.CV↗
arXiv2024

Compactness of composition operators on the Bergman space of the bidisc

Let $\varphi$ be a holomorphic self map of the bidisc that is Lipschitz on the closure. We show that the composition operator $C_{\varphi}$ is compact on the Bergman space if and only if $\varphi(\overline{\mathbb{D}^2})\cap \mathbb{T}^2=\emptyset$ and $\varphi(\overline{\mathbb{D}^2}\setminus \mathbb{T}^2)\cap b\mathbb{D}^2=\emptyset$.

math.CV↗
arXiv2024

A geometric condition for the invertibility of Toeplitz operators on the Bergman space

Invertibility of Toeplitz operators on the Bergman space and the related Douglas problem are long standing open problems. In this paper we study the invertibility problem under the novel geometric condition on the image of the symbols, which relaxes the standard positivity condition. We show that under our geometric assumption, the Toeplitz operator $T_\varphi$ is invertible if and only if the Berezin transform of $|\varphi|$ is invertible in $L^{\infty}$. It is well known that the Douglas problem is still open for harmonic functions. We study a class of rather general harmonic polynomials and characterize the invertibility of the corresponding Toeplitz operators. We also give a number of related results and examples.

math.FA↗
arXiv2022

On spectra of Hankel operators on the polydisc

We give sufficient conditions for the essential spectrum of the Hermitian square of a class of Hankel operators on the Bergman space of the polydisc to contain intervals. We also compute the spectrum in case the symbol is a monomial.

math.FA↗
arXiv2021

Projected composition operators on pseudoconvex domains

Let $Ω\subset \mathbb{C}^n$ be a smooth bounded pseudoconvex domain and $A^2 (Ω)$ denote its Bergman space. Let $P:L^2(Ω)\longrightarrow A^2(Ω)$ be the Bergman projection. For a measurable $φ:Ω\longrightarrow Ω$, the projected composition operator is defined by $(K_φf)(z) = P(f \circ φ)(z), z \inΩ, f\in A^2 (Ω).$ In 1994, Rochberg studied boundedness of $K_φ$ on the Hardy space of the unit disk and obtained different necessary or sufficient conditions for boundedness of $K_φ$. In this paper we are interested in projected composition operators on Bergman spaces on pseudoconvex domains. We study boundedness of this operator under the smoothness assumptions on the symbol $φ$ on $\overlineΩ$.

math.CV↗
arXiv2021

Berezin regularity of domains in C^n and the essential norms of Toeplitz operators

For the open unit disc $\mathbb{D}$ in the complex plane, it is well known that if $ϕ\in C(\overline{\mathbb{D}})$ then its Berezin transform $\widetildeϕ$ also belongs to $C(\overline{\mathbb{D}})$. We say that $\mathbb{D}$ is BC-regular. In this paper we study BC-regularity of some pseudoconvex domains in $\mathbb{C}^n$ and show that the boundary geometry plays an important role. We also establish a relationship between the essential norm of an operator in a natural Toeplitz subalgebra and its Berezin transform.

math.CV↗
arXiv2020

Zero products of Toeplitz operators on Reinhardt domains

Let $\Omega$ be a bounded Reinhardt domain in $\mathbb{C}^n$ and $\phi_1,\ldots,\phi_m$ be finite sums of bounded quasi-homogeneous functions. We show that if the product of Toeplitz operators $T_{\phi_m}\cdots T_{\phi_1}=0$ on the Bergman space on $\Omega$, then $\phi_j=0$ for some $j$.

math.CV↗
arXiv2018

Determinantal hypersurfaces and representations of Coxeter groups

Given a finite generating set $T=\{g_0,\dots, g_n\}$ of a group $G$, and a representation $ρ$ of $G$ on a Hilbert space $V$, we investigate how the geometry of the set $D(T,ρ)=\{ [x_0 : \dots : x_n] \in\mathbb C\mathbb P^n \mid \sum x_iρ(g_i) \text{ not invertible} \}$ reflects the properties of $ρ$. When $V$ is finite-dimensional this is an algebraic hypersurface in $\mathbb C\mathbb P^n$. In the special case $T=G$ and $ρ=$ the left regular representation of $G$, this hypersurface is defined by the \emph{group determinant}, an object studied extensively in the founding work of Frobenius that lead to the creation of representation theory. We focus on the classic case when $G$ is a finite Coxeter group, and make $T$ by adding the identity element $1_G$ to a Coxeter generating set for $G$. Under these assumptions we show in our first main result that if $ρ$ is the left regular representation, then $D(T,ρ)$ determines the isomorphism class of $G$. Our second main result is that if $G$ is not of exceptional type, and $ρ$ is any finite dimensional representation, then $D(T,ρ)$ determines $ρ$.

math.RT↗
arXiv2018

Essential norm estimates for Hankel operators on convex domains in $\mathbb{C}^2$

Let $Ω\subset \mathbb{C}^2$ be a bounded convex domain with $C^1$-smooth boundary and $φ\in C^1(\overlineΩ)$ such that $φ$ is harmonic on the nontrivial disks in the boundary. We estimate the essential norm of the Hankel operator $H_φ$ in terms of the $\overline{\partial}$ derivatives of $φ$ "along" the nontrivial disks in the boundary.

math.CV↗
arXiv2018

The lattices of invariant subspaces of a class of operators on the Hardy space

In the authors' first paper, Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. Current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator.

math.CV↗
arXiv2017

A local weighted Axler-Zheng theorem in $\mathbb{C}^n$

The well-known Axler-Zheng theorem characterizes compactness of finite sums of finite products of Toeplitz operators on the unit disk in terms of the Berezin transform of these operators. Subsequently this theorem was generalized to other domains and appeared in different forms, including domains in $\mathbb{C}^n$ on which the $\overline{\partial}$-Neumann operator $N$ is compact. In this work we remove the assumption on $N$, and we study weighted Bergman spaces on smooth bounded pseudoconvex domains. We prove a local version of the Axler-Zheng theorem characterizing compactness of Toeplitz operators in the algebra generated by symbols continuous up to the boundary in terms of the behavior of the Berezin transform at strongly pseudoconvex points. We employ a Forelli-Rudin type inflation method to handle the weights.

math.CV↗
arXiv2016

A New Necessary Condition for the Hyponormality of Toeplitz Operators on the Bergman Space

A well known result of C. Cowen states that, for a symbol $\varphi \in L^{\infty }, \; \varphi \equiv \bar{f}+g \;\;(f,g\in H^{2})$, the Toeplitz operator $T_{\varphi }$ acting on the Hardy space of the unit circle is hyponormal if and only if $f=c+T_{\bar{h}}g,$ for some $c\in {\mathbb C}$, $h\in H^{\infty }$, $\left\| h\right\| _{\infty}\leq 1.$ \ In this note we consider possible versions of this result in the {\it Bergman} space case. \ Concretely, we consider Toeplitz operators on the Bergman space of the unit disk, with symbols of the form $$\varphi \equiv \alpha z^n+\beta z^m +\gamma \overline z ^p + \delta \overline z ^q,$$ where $\alpha, \beta, \gamma, \delta \in \mathbb{C}$ and $m,n,p,q \in \mathbb{Z}_+$, $m < n$ and $p < q$. \ By letting $T_{\varphi}$ act on vectors of the form $$z^k+c z^{\ell}+d z^r \; \; (k<\ell<r),$$ we study the asymptotic behavior of a suitable matrix of inner products, as $k \rightarrow \infty$. \ As a result, we obtain a sharp inequality involving the above mentioned data: $$ \left|\alpha \right|^2 n^2 + \left|\beta \right|^2 m^2 - \left|\gamma \right|^2 p^2 - \left|\delta \right|^2 q^2 \ge 2 \left|\bar \alpha \beta m n - \bar \gamma \delta p q \right|. $$ This inequality improves a number of existing results, and it is intended to be a precursor of basic necessary conditions for joint hyponormality of tuples of Toeplitz operators acting on Bergman spaces in one or several complex variables.

math.FA↗
arXiv2015

Mapping Properties of Weighted Bergman Projection Operators on Reinhardt Domains

We show that on smooth complete Reinhardt domains, weighted Bergman projection operators corresponding to exponentially decaying weights are unbounded on $L^p$ spaces for all $p\not=2$. On the other hand, we also show that the exponentially weighted projection operators are bounded on Sobolev spaces on the unit ball.

math.CV↗
arXiv2015

Adjoints of linear fractional composition operators on weighted Hardy spaces

It is well known that on the Hardy space $H^2(\mathbb{D})$ or weighted Bergman space $A^2_α(\mathbb{D})$ over the unit disk, the adjoint of a linear fractional composition operator equals the product of a composition operator and two Toeplitz operators. On $S^2(\mathbb{D})$, the space of analytic functions on the disk whose first derivatives belong to $H^2(\mathbb{D})$, Heller showed that a similar formula holds modulo the ideal of compact operators. In this paper we investigate what the situation is like on other weighted Hardy spaces.

math.FA↗
arXiv2013

Toeplitzness of composition operators in several variables

Motivated by the work of Nazarov and Shapiro on the unit disk, we study asymptotic Toeplitzness of composition operators on the Hardy space of the unit sphere in C^n. We extend some of their results but we also show that new phenomena appear in higher dimensions.

math.FA↗