Recent zbMATH articles in MSC 32A36https://zbmath.org/atom/cc/32A362021-07-26T21:45:41.944397ZWerkzeugCompactness of operators on the Bergman space of the Thullen domainhttps://zbmath.org/1463.320062021-07-26T21:45:41.944397Z"Huo, Zhenghui"https://zbmath.org/authors/?q=ai:huo.zhenghui"Wick, Brett D."https://zbmath.org/authors/?q=ai:wick.brett-dLet \(\mathcal{U}^\alpha = \{ z = (z_1,z_2) \in \mathbb{C}^2 :\vert z_1\vert^ {2/\alpha} + \vert z_2\vert^2 <1 \}\), where \(\alpha >0, \alpha \neq 1.\) Thullen showed that holomorphic automorphisms \(\varphi \) on \(\mathcal{U}^\alpha \) are of the form : \(\varphi (z_1,z_2) = \left (e^{i \theta_1} z_1 \left (\frac{\sqrt {1:\vert w\vert^2}}{1-z_2\bar w} \right)^\alpha, e^{i \theta_2} \frac{w-z_2}{1-z_2 \bar w} \right),\) where \(\vert w\vert<1\) and \(\theta_1, \theta_2 \in \mathbb{R}.\) The authors first give a sufficient condition for the boundedness of an operator on the Bergman space \(A^2(\mathcal{U}^\alpha)\) whose adjoint and itself are defined a priori only on the linear span of the normalized reproducing kernels \(\{k_z\}.\) Then they show that a Toeplitz operator with an \(L^\infty\) symbol is compact if and only if \(\lim_{d(z,0) \to 1} \vert Tk_z\vert = 0,\) where \(d(.,.)\) denotes the Skwarczyński distance \(d(z,w) = (1- \vert\langle k_z, k_w \rangle \vert)^{1/2}.\)On a class of inner functions in a half-spacehttps://zbmath.org/1463.320072021-07-26T21:45:41.944397Z"Shamoyan, Faĭzo Agitovich"https://zbmath.org/authors/?q=ai:shamoyan.faizo-aSummary: In the paper we obtain necessary and sufficient conditions for the weight vector function, under which a given inner function is weakly invertible in the weighted functions of holomorphic functions in a tubular domain.Weighted composition operators on weighted Bergman spaces and weighted Bloch spaceshttps://zbmath.org/1463.470792021-07-26T21:45:41.944397Z"Hassanlou, Mostafa"https://zbmath.org/authors/?q=ai:hassanlou.mostafa"Vaezi, Hamid"https://zbmath.org/authors/?q=ai:vaezi.hamidSummary: In this paper, we characterize the boundedness and compactness of weighted composition operators from weighted Bergman spaces to weighted Bloch spaces. Also, we investigate weighted composition operators on weighted Bergman spaces and extend the obtained results in the unit ball of \(\mathbb{C}^n\).Toeplitz operators with \(L^1\) symbols on the weighted Bergman spaceshttps://zbmath.org/1463.470912021-07-26T21:45:41.944397Z"He, Zhonghua"https://zbmath.org/authors/?q=ai:he.zhonghua"Cao, Guangfu"https://zbmath.org/authors/?q=ai:cao.guangfuSummary: This paper characterizes the boundedness and compactness of a Toeplitz operator on the weighted Bergman space with a \(L^1\) symbol. This result extends known results in the cases when the symbol is either a positive \(L^1\) function, an \(L^\infty\) function or a general \(BMO^1\) function. In addition, we also give some estimates about the norm and essential norm of Toeplitz operators with \(L^1\) symbols.Products and commuting of Toeplitz operators on the Bergman space of the polydiskhttps://zbmath.org/1463.470932021-07-26T21:45:41.944397Z"Sun, Zhiling"https://zbmath.org/authors/?q=ai:sun.zhilingSummary: In this paper, we give some necessary and sufficient conditions for the product of two Toeplitz operators with certain symbols to be a Toeplitz operator and give a formula for the symbol of the product on the Bergman space of polydisk. Next, the corresponding commuting problem of Toeplitz operator is studied. All the results are stated in terms of the Mellin transform of the symbol.Extended Cesàro operators from weighted Bergman space to Bloch-type spaces on the unit polydiskhttps://zbmath.org/1463.471422021-07-26T21:45:41.944397Z"Zhao, Yanhui"https://zbmath.org/authors/?q=ai:zhao.yanhui"Liao, Chunyan"https://zbmath.org/authors/?q=ai:liao.chunyan"Deng, Chunhong"https://zbmath.org/authors/?q=ai:deng.chunhong"Wu, Xiuyun"https://zbmath.org/authors/?q=ai:wu.xiuyunSummary: Some questions of extended Cesàro operators from weighted Bergman space to Bloch-type spaces on the unit polydisk were studied. By applying the methods of functional analysis and several complex variables, the necessary and sufficient conditions were given for extended Cesàro operators to be bounded and compact from weighted Bergman space to Bloch-type spaces on the unit polydisk and on the unit disk.Extended Cesàro operator from weighted Bergman spaces to \({\mathcal{Z}_\mu}\) type spaces on the unit ballhttps://zbmath.org/1463.471432021-07-26T21:45:41.944397Z"Zhao, Yanhui"https://zbmath.org/authors/?q=ai:zhao.yanhui"Wu, Xiuyun"https://zbmath.org/authors/?q=ai:wu.xiuyun"Liao, Chunyan"https://zbmath.org/authors/?q=ai:liao.chunyanSummary: Some questions of extended Cesàro operator from weighted Bergman space to \({\mathcal{Z}_\mu}\) type spaces in the unit ball were studied in this paper. By the methods of functional analysis and several complex variables, the necessary and sufficient conditions are given for extended Cesàro operator to be bounded and compact from weighted Bergman space to \({\mathcal{Z}_\mu}\) type spaces in the unit ball.