In this paper, we focus on the boundedness and compactness of the Toeplitz operators and the Hankel operators on the $\mathcal{D}$-weighted Bergman spaces $A_{\omega}^p$ ($1\leq p<\infty$). In particular, the boundedness of Toeplitz operators with $\mathrm{BMO}_\omega^p$-symbols on $A_{\omega}^p$ is characterized in terms of the Berezin-type transform. Several sufficient conditions for the Toeplitz operators $T_f^\omega$ with locally integrable symbols to be bounded (resp. compact) for each $1\leq p<\infty$ are presented. Moreover, sufficient and necessary conditions for the boundedness of the Toeplitz operator $T_{\bar{f}}^\omega:A_\omega^1\rightarrow A_\omega^1$ and the Hankel operator $H_{\bar{f}}^\omega:A_\omega^1\rightarrow L_\omega^1$ with co-analytic symbol are established.
Let $\omega$ be a radial $\widehat{\mathcal D}$-weight and $u$ be a bounded function on the unit disk $\mathbb D$. We prove that the Toeplitz operator \(T_{\omega,u}\) is compact on \(A_\omega^2\) if and only if its Berezin transform vanishes at the boundary. Our approach is based on a polynomial frame for $A_\omega^2$...
We investigate the hypercyclicity of Toeplitz operators on the Bergman space $L_{A}^{2}(\mathbb{D})$ with symbols of the form $\Psi(z) = \gamma\bar{z}+\psi(z)$, where $\gamma \in \mathbb{C} \setminus \{0\}$ and $\psi$ is analytic on an open neighborhood of the closed unit disc $\overline{\mathbb{D}}$. Our approach bypa...
Let $\varphi\in L^\infty(\D)$. We study compactness criteria for \(T_\varphi\) on the Bergman space $A^2(\D)$. Axler and Zheng~\cite{AZ1998} established a necessary and sufficient condition for compactness in terms of the Berezin transform. For a general bounded measurable function $\varphi$, however, its Berezin trans...
Let $L=-\Delta+V$ be a Schr\"odinger operator on $\mathbb{R}^n$, where $\Delta$ is the Laplacian and $V$ satisfies the reverse H\"older inequality ${\rm RH}_q$ for some $q>n/2$. In this paper, we study the behavior of the Littlewood--Paley operators $s_L$ and $S_L$, as well as the semigroup maximal operator $T^*_L$, on...
The pentablock is a domain in $\mathbb C^3$ arising in the study of the $\mu$-synthesis problem and is defined by \[ \mathcal P = \left\{ (a_{21},\operatorname{tr}(A),\det(A)): A=[a_{ij}]_{i,j=1}^2\in M_2(\mathbb C),\ \|A\|<1 \right\}. \] We introduce the Hardy space $H^2(\mathcal P)$ associated with the canonical meas...
Let $L=-\Delta_k+V$ be a Schr\"odinger operator associated with the Dunkl Laplacian and a nonnegative potential. We characterize the Hardy space $H_L^1$, defined by the maximal operator of the semigroup generated by $-L$, in terms of the Riesz transforms associated with $L$. Under a local integral condition on the pote...
Agnieszka Hejna-Łyżwa· 0 citations
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