arXiv subjects

Sonmez Sahutoglu

Publications and source records attributed to Sonmez Sahutoglu.

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

Compactness of composition operators on the Bergman space of the bidisc

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

math.CV↗

arXiv2025

Compactness of Hankel and Toeplitz operators on convex Reinhardt domains in $\mathbb{C}^2$

We study compactness of Hankel and Toeplitz operators on Bergman spaces of convex Reinhardt domains in $\mathbb{C}^2$ and we restrict the symbols to the class of functions that are continuous on the closure of the domain. We prove that Toeplitz operators as well as the Hermitian squares of Hankel operators are compact if and only if the Berezin transforms of the operators vanish on the boundary of the domain.

math.CV↗

arXiv2024

On compactness of products of Toeplitz operators

We study compactness of product of Toeplitz operators with symbols continuous on the closure of the polydisc in terms of behavior of the symbols on the boundary. For certain classes of symbols $f$ and $g$, we show that $T_fT_g$ is compact if and only if $fg$ vanishes on the boundary. We provide examples to show that for more general symbols, the vanishing of $fg$ on the whole polydisc might not imply the compactness of $T_fT_g$. On the other hand, the reverse direction is closely related to the zero product problem for Toeplitz operators on the unit disc, which is still open.

math.FA↗

arXiv2024

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↗

arXiv2024

Compactness of Toeplitz operators with continuous symbols on pseudoconvex domains in $\mathbb{C}^n$

Let $Ω$ be a bounded pseudoconvex domain in $\mathbb{C}^n$ with Lipschitz boundary and $ϕ$ be a continuous function on $\overlineΩ$. We show that the Toeplitz operator $T_ϕ$ with symbol $ϕ$ is compact on the weighted Bergman space if and only if $ϕ$ vanishes on the boundary of $Ω$. We also show that compactness of the Toeplitz operator $T^{p,q}_ϕ$ on $\overline{\partial}$-closed $(p,q)$-forms for $0\leq p\leq n$ and $q\geq 1$ is equivalent to $ϕ=0$ on $Ω$.

math.CV↗

arXiv2022

A sufficient condition for $L^p$ regularity of the Berezin transform

We prove that the Berezin transform is $L^p$ regular on a large class of domains in $\mathbb{C}^n$ and not $L^2$ regular on the Hartogs triangle.

math.CV↗

arXiv2021

A Sufficient condition for compactness of Hankel operators

Let $Ω$ be a bounded convex domain in $\mathbb{C}^{n}$. We show that if $φ\in C^{1}(\overlineΩ)$ is holomorphic along analytic varieties in $bΩ$, then $H^{q}_φ$, the Hankel operator with symbol $φ$, is compact. We have shown the converse earlier, so that we obtain a characterization of compactness of these operators in terms of the behavior of the symbol relative to analytic structure in the boundary. A corollary is that Toeplitz operators with these symbols are Fredholm (of index zero).

math.CV↗

arXiv2021

On compactness and $L^p$-regularity in the $\overline{\partial}$-Neumann problem

Let $Ω$ be a $C^4$-smooth bounded pseudoconvex domain in $\mathbb{C}^2$. We show that if the $\overline{\partial}$-Neumann operator $N_1$ is compact on $L^2_{(0,1)}(Ω)$ then the embedding operator $\mathcal{J}:Dom(\overline{\partial})\cap Dom(\overline{\partial}^*) \to L^2_{(0,1)}(Ω)$ is $L^p$-regular for all $2\leq p<\infty$.

math.CV↗

arXiv2021

Zero products of Toeplitz operators on Reinhardt domains

Let $Ω$ be a bounded Reinhardt domain in $\mathbb{C}^n$ and $ϕ_1,\ldots,ϕ_m$ be finite sums of bounded quasi-homogeneous functions. We show that if the product of Toeplitz operators $T_{ϕ_m}\cdots T_{ϕ_1}=0$ on the Bergman space on $Ω$, then $ϕ_j=0$ for some $j$.

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

Compactness of Hankel operators with continuous symbols on convex domains

Let $Ω$ be a bounded convex domain in $\mathbb{C}^{n}$, $n\geq 2$, $1\leq q\leq (n-1)$, and $ϕ\in C(\barΩ)$. If the Hankel operator $H^{q-1}_ϕ$ on $(0,q-1)$--forms with symbol $ϕ$ is compact, then $ϕ$ is holomorphic along $q$--dimensional analytic (actually, affine) varieties in the boundary. We also prove a partial converse: if the boundary contains only `finitely many' varieties, $1\leq q\leq n$, and $ϕ\in C(\barΩ)$ is analytic along the ones of dimension $q$ (or higher), then $H^{q-1}_ϕ$ is compact.

math.CV↗

arXiv2020

On convergence of the Berezin transforms

We prove approximation results about sequences of Berezin transforms of finite sums of finite product of Toeplitz operators (and bounded linear maps, in general) in the spirit of Ramadanov and Skwarczynski theorems that are about convergence of Bergman kernels.

math.CV↗

arXiv2019

Convex domains, Hankel operators, and maximal estimates

Let $1\leq q\leq (n-1)$. We first show that a necessary condition for a Hankel operator on $(0,q-1)$-forms on a convex domain to be compact is that its symbol is holomorphic along $q$-dimensional analytic varieties in the boundary. Because maximal estimates (equivalently, a comparable eigenvalues condition on the Levi form of the boundary) turn out to be favorable for compactness of Hankel operators, this result then implies that on a convex domain, maximal estimates exclude analytic varieties from the boundary, except ones of top dimension $(n-1)$ (and their subvarieties). Some of our techniques apply to general pseudoconvex domains to show that if the Levi form has comparable eigenvalues, or equivalently, if the domain admits maximal estimates, then compactness and subellipticity hold for forms at some level $q$ if and only if they hold at all levels.

math.CV↗

arXiv2018

Compactness of Hankel operators with symbols continuous on the closure of pseudoconvex domains

Let $Ω$ be a bounded pseudoconvex domain in $\mathbb{C}^2$ with Lipschitz boundary or a bounded convex domain in $\mathbb{C}^n$ and $ϕ\in C(\overlineΩ)$ such that $H_ϕ$ is compact on $A^2(Ω)$. Then $ϕ\circ f$ is holomorphic for any holomorphic $f:\mathbb{D}\rightarrow bΩ$.

math.CV↗

arXiv2018

The restriction operator on Bergman spaces

We study the restriction operator from the Bergman space of a domain in $\mathbb{C}^n$ to the Bergman space of a non-empty open subset of the domain. We relate the restriction operator to the Toeplitz operator on the Bergman space of the domain whose symbol is the characteristic function of the subset. Using the biholomorphic invariance of the spectrum of the associated Toeplitz operator, we study the restriction operator from the Bergman space of the unit disc to the Bergman space of subdomains with large symmetry groups, such as horodiscs and subdomains bounded by hypercycles. Furthermore, we prove a sharp estimate of the norm of the restriction operator in case the domain and the subdomain are balls. We also study various operator theoretic properties of the restriction operator such as compactness and essential norm estimates.

math.CV↗

arXiv2018

On a theorem of Bishop and commutants of Toeplitz operators in $\mathbb{C}^n$

We prove an approximation theorem on a class of domains in $\mathbb{C}^n$ on which the $\overline{\partial}$-problem is solvable in $L^{\infty}$. Furthermore, as a corollary, we obtain a version of the Axler-Čučković-Rao Theorem in higher dimensions.

math.CV↗

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↗