Recent zbMATH articles in MSC 30Hhttps://www.zbmath.org/atom/cc/30H2022-05-16T20:40:13.078697ZWerkzeugRiesz projection and bounded mean oscillation for Dirichlet serieshttps://www.zbmath.org/1483.300092022-05-16T20:40:13.078697Z"Konyagin, Sergei"https://www.zbmath.org/authors/?q=ai:konyagin.sergey-v"Queffélec, Hervé"https://www.zbmath.org/authors/?q=ai:queffelec.herve"Saksman, Eero"https://www.zbmath.org/authors/?q=ai:saksman.eero"Seip, Kristian"https://www.zbmath.org/authors/?q=ai:seip.kristianSummary: We prove that the norm of the Riesz projection from \(L^\infty (\mathbb{T}^n)\) to \(L^p(\mathbb{T}^n)\) is \(1\) for all \(n\ge 1\) only if \(p\le 2\), thus solving a problem posed by \textit{J. Marzo} and the fourth author [Bull. Sci. Math. 135, No. 3, 324--331 (2011; Zbl 1221.42013)]. This shows that \(H^p(\mathbb{T}^{\infty})\) does not contain the dual space of \(H^1(\mathbb{T}^{\infty})\) for any \(p > 2\). We then note that the dual of \(H^1(\mathbb{T}^{\infty})\) contains, via the Bohr lift, the space of Dirichlet series in BMOA of the right half-plane. We give several conditions showing how this BMOA space relates to other spaces of Dirichlet series. Finally, relating the partial sum operator for Dirichlet series to Riesz projection on \(\mathbb{T}\), we compute its \(L^p\) norm when \(1 < p < \infty\), and we use this result to show that the \(L^\infty\) norm of the \(N\) th partial sum of a bounded Dirichlet series over \(d\)-smooth numbers is of order \(\log\log N\).Paatero's \(V(k)\) space and a claim by Pinchukhttps://www.zbmath.org/1483.300342022-05-16T20:40:13.078697Z"Andreev, Valentin V."https://www.zbmath.org/authors/?q=ai:andreev.valentin-v"Bekker, Miron B."https://www.zbmath.org/authors/?q=ai:bekker.miron-b"Cima, Joseph A."https://www.zbmath.org/authors/?q=ai:cima.joseph-aSummary: In this article we obtain a factorization theorem for the functions in Paatero's \(V(k)\) space. We bring attention to a significant result of Pinchuk which unfortunately is false. This result relates measures associated to functions in \(V(k)\) and an integral representation theorem for such functions. We prove necessary and sufficient conditions for a wide class of functions (in particular, the polynomials) to belong to the Paatero class based on the geometry of their critical points, and obtain explicit representation of the measures associated to a wide class of such polynomials that includes the Suffridge polynomials.The use of the isometry of function spaces with different numbers of variables in the theory of approximation of functionshttps://www.zbmath.org/1483.300582022-05-16T20:40:13.078697Z"Bushev, D. M."https://www.zbmath.org/authors/?q=ai:bushev.d-m"Abdullayev, F. G."https://www.zbmath.org/authors/?q=ai:abdullayev.fahreddin-g"Kal'chuk, I. V."https://www.zbmath.org/authors/?q=ai:kalchuk.inna-v"Imashkyzy, M."https://www.zbmath.org/authors/?q=ai:imashkyzy.meerimSummary: In the work, we found integral representations for function spaces that are isometric to spaces of entire functions of exponential type, which are necessary for giving examples of equality of approximation characteristics in function spaces isometric to spaces of non-periodic functions. Sufficient conditions are obtained for the nonnegativity of the delta-like kernels used to construct isometric function spaces with various numbers of variables.Approximation of functions and all derivatives on compact setshttps://www.zbmath.org/1483.300682022-05-16T20:40:13.078697Z"Armeniakos, Sotiris"https://www.zbmath.org/authors/?q=ai:armeniakos.sotiris"Kotsovolis, Giorgos"https://www.zbmath.org/authors/?q=ai:kotsovolis.giorgos"Nestoridis, Vassili"https://www.zbmath.org/authors/?q=ai:nestoridis.vassiliSummary: In Mergelyan type approximation we uniformly approximate functions on compact sets \(K\) by polynomials or rational functions or holomorphic functions on varying open sets containing \(K\). In the present paper we consider analogous approximation, where uniform convergence on \(K\) is replaced by uniform approximation on \(K\) of all order derivatives.On \(s\)-extremal Riemann surfaces of even genushttps://www.zbmath.org/1483.300792022-05-16T20:40:13.078697Z"Kozłowska-Walania, Ewa"https://www.zbmath.org/authors/?q=ai:kozlowska-walania.ewaSummary: We consider Riemann surfaces of even genus \(g\) with the action of the group \(\mathcal{D}_n\times \mathbb{Z}_2\), with \(n\) even. These surfaces have the maximal number of 4 non-conjugate symmetries and shall be called \textit{\(s\)-extremal}. We show various results for such surfaces, concerning the total number of ovals, topological types of symmetries, hyperellipticity degree and the minimal genus problem. If in addition an \(s\)-extremal Riemann surface has the maximal total number of ovals, then it shall simply be called \textit{extremal}. In the main result of the paper we find all the families of extremal Riemann surfaces of even genera, depending on if one of the symmetries is fixed-point free or not.Bounded extremal problems in Bergman and Bergman-Vekua spaceshttps://www.zbmath.org/1483.300892022-05-16T20:40:13.078697Z"Delgado, Briceyda B."https://www.zbmath.org/authors/?q=ai:delgado.briceyda-b"Leblond, Juliette"https://www.zbmath.org/authors/?q=ai:leblond.julietteSummary: We analyze Bergman spaces \(A_f^p(\mathbb{D})\) of generalized analytic functions of solutions to the Vekua equation \(\bar{\partial}w = (\bar{\partial}f/f)\bar{w}\) in the unit disc of the complex plane, for Lipschitz-smooth non-vanishing real valued functions \(f\) and \(1<p<\infty\). We consider a family of bounded extremal problems (best constrained approximation) in the Bergman space \(A^p(\mathbb{D})\) and in its generalized version \(A^p_f(\mathbb{D})\), that consists in approximating a function in subsets of \(\mathbb{D}\) by the restriction of a function belonging to \(A^p(\mathbb{D})\) or \(A^p_f(\mathbb{D})\) subject to a norm constraint. Preliminary constructive results are provided for \(p = 2\).Variability regions for the second derivative of bounded analytic functionshttps://www.zbmath.org/1483.300942022-05-16T20:40:13.078697Z"Chen, Gangqiang"https://www.zbmath.org/authors/?q=ai:chen.gangqiang"Yanagihara, Hiroshi"https://www.zbmath.org/authors/?q=ai:yanagihara.hiroshiSummary: Let \(z_0\) and \(w_0\) be given points in the open unit disk \({\mathbb{D}}\) with \(|w_0| < |z_0|\). Let \({\mathcal{H}}_0\) be the class of all analytic self-maps \(f\) of \({\mathbb{D}}\) normalized by \(f(0)=0\), and \({\mathcal{H}}_0 (z_0,w_0) = \{ f \in{\mathcal{H}}_0 : f(z_0) =w_0\} \). In this paper, we explicitly determine the variability region of \(f''(z_0)\) when \(f\) ranges over \({\mathcal{H}}_0 (z_0,w_0)\). Moreover, we approximate this region numerically in some special cases, to illustrate our main result.Corona theorem for the Dirichlet-type spacehttps://www.zbmath.org/1483.300952022-05-16T20:40:13.078697Z"Luo, Shuaibing"https://www.zbmath.org/authors/?q=ai:luo.shuaibingSummary: This paper utilizes Cauchy's transform and duality for the Dirichlet-type space \(D(\mu)\) with positive superharmonic weight \(U_{\mu}\) on the unit disk \(\mathbb{D}\) to establish the corona theorem for the Dirichlet-type multiplier algebra \(M\big( D(\mu)\big)\) that: if
\[
\{ f_1,\ldots ,f_n\} \subseteq M\big( D(\mu )\big) \quad \text{and}\quad \inf_{z\in \mathbb{D}}\sum_{j=1}^n |f_j (z)|>0
\]
then
\[
\exists \,\{ g_1,\ldots,g_n\}\subseteq M\big( D(\mu)\big) \quad \text{such that}\quad \sum_{j=1}^n f_j g_j =1,
\]
thereby generalizing \textit{L. Carleson}'s corona theorem for \(M(H^2)=H^{\infty}\) in [Ann. Math. (2) 76, 547--559 (1962; Zbl 0112.29702)] and \textit{J.Xiao}'s corona theorem for \(M(\mathscr{D})\subset H^{\infty}\) in [Manuscr. Math. 97, No. 2, 217--232 (1998; Zbl 1049.30025)] thanks to
\[
D(\mu )=
\begin{cases}
\text{Hardy space } H^2 \quad & \text{as}\quad \text{d}\mu (z)=(1-|z|^2)\,\text{d}A(z)\quad \forall z\in \mathbb{D}; \\
\text{Dirichlet space}\; \mathscr{D} & \text{as}\quad \text{d}\mu (z)=|\text{d}z|\quad \forall z\in \mathbb{T} =\partial\mathbb{D}. \end{cases}
\]Composition of analytic paraproductshttps://www.zbmath.org/1483.300962022-05-16T20:40:13.078697Z"Aleman, Alexandru"https://www.zbmath.org/authors/?q=ai:aleman.alexandru"Cascante, Carme"https://www.zbmath.org/authors/?q=ai:cascante.carme"Fàbrega, Joan"https://www.zbmath.org/authors/?q=ai:fabrega.joan"Pascuas, Daniel"https://www.zbmath.org/authors/?q=ai:pascuas.daniel"Peláez, José Ángel"https://www.zbmath.org/authors/?q=ai:pelaez.jose-angelSummary: For a fixed analytic function \(g\) on the unit disc \(\mathbb{D}\), we consider the analytic paraproducts induced by \(g\), which are defined by \(T_g f(z)=\int_0^z f(\zeta)g^\prime(\zeta)d\zeta\), \(S_g f(z)=\int_0^z f^\prime(\zeta) g(\zeta)d\zeta\), and \(M_g f(z)=f(z) g(z)\). The boundedness of these operators on various spaces of analytic functions on \(\mathbb{D}\) is well understood. The original motivation for this work is to understand the boundedness of compositions of two of these operators, for example \(T_g^2\), \(T_gS_g\), \(M_g T_g\), etc. Our methods yield a characterization of the boundedness of a large class of operators contained in the algebra generated by these analytic paraproducts acting on the classical weighted Bergman and Hardy spaces in terms of the symbol \(g\). In some cases it turns out that this property is not affected by cancellation, while in others it requires stronger and more subtle restrictions on the oscillation of the symbol \(g\) than the case of a single paraproduct.A note on the phase retrieval of holomorphic functionshttps://www.zbmath.org/1483.300972022-05-16T20:40:13.078697Z"Perez, Rolando III"https://www.zbmath.org/authors/?q=ai:perez.rolando-iiiSummary: We prove that if \(f\) and \(g\) are holomorphic functions on an open connected domain, with the same moduli on two intersecting segments, then \(f=g\) up to the multiplication of a unimodular constant, provided the segments make an angle that is an irrational multiple of \(\pi\). We also prove that if \(f\) and \(g\) are functions in the Nevanlinna class, and if \(|f|=|g|\) on the unit circle and on a circle inside the unit disc, then \(f=g\) up to the multiplication of a unimodular constant.Norms of inclusions between some spaces of analytic functionshttps://www.zbmath.org/1483.300982022-05-16T20:40:13.078697Z"Llinares, Adrián"https://www.zbmath.org/authors/?q=ai:llinares.adrianSummary: The inclusions between the Besov spaces \(B^q\), the Bloch space \(\mathcal{B}\) and the standard weighted Bergman spaces \(A^p_{\alpha}\) are completely understood, but the norms of the corresponding inclusion operators are in general unknown. In this work, we compute or estimate asymptotically the norms of these inclusions.Complete interpolating sequences for small Fock spaceshttps://www.zbmath.org/1483.300992022-05-16T20:40:13.078697Z"Omari, Youssef"https://www.zbmath.org/authors/?q=ai:omari.youssefSummary: We give a characterization of complete interpolating sequences for the Fock spaces \(\mathcal{F}_\varphi^p\), \(1\leq p< \infty\), where \(\varphi(z)=\alpha(\log^+|z|)^2\), \(\alpha > 0\). Our results are analogous to the classical Kadets-Ingham's 1/4-Theorem on perturbation of Riesz bases of complex exponentials, and they answer a question asked by \textit{A. Baranov} et al. [J. Math. Pures Appl. (9) 103, No. 6, 1358--1389 (2015; Zbl 1315.30009)].Extreme points and support points of families of harmonic Bloch mappingshttps://www.zbmath.org/1483.310022022-05-16T20:40:13.078697Z"Deng, Hua"https://www.zbmath.org/authors/?q=ai:deng.hua"Ponnusamy, Saminathan"https://www.zbmath.org/authors/?q=ai:ponnusamy.saminathan"Qiao, Jinjing"https://www.zbmath.org/authors/?q=ai:qiao.jinjingSummary: In this paper, the main aim is to discuss the existence of the extreme points and support points of families of harmonic Bloch mappings and little harmonic Bloch mappings. First, in terms of the Bloch unit-valued set, we prove a necessary condition for a harmonic Bloch mapping (resp. a little harmonic Bloch mapping) to be an extreme point of the unit ball of the normalized harmonic Bloch spaces (resp. the normalized little harmonic Bloch spaces) in the unit disk \(\mathbb{D}\). Then we show that a harmonic Bloch mapping \(f\) is a support point of the unit ball of the normalized harmonic Bloch spaces in \(\mathbb{D}\) if and only if the Bloch unit-valued set of \(f\) is not empty. We also give a characterization for the support points of the unit ball of the harmonic Bloch spaces in \(\mathbb{D}\).Corona theorem for strictly pseudoconvex domainshttps://www.zbmath.org/1483.320162022-05-16T20:40:13.078697Z"Gwizdek, Sebastian"https://www.zbmath.org/authors/?q=ai:gwizdek.sebastianSummary: Nearly 60 years have passed since Lennart Carleson gave his proof of Corona Theorem for unit disc in the complex plane. It was only recently that \textit{M. Kosiek} and \textit{K. Rudol} [``Corona theorem'', Preprint, \url{arXiv:2106.15683}] obtained the first positive result for Corona Theorem in multidimensional case. Using duality methods for uniform algebras the authors proved ``abstract'' Corona Theorem which allowed to solve Corona Problem for a wide class of regular domains. In this paper we expand Corona Theorem to strictly pseudoconvex domains with smooth boundaries.Riesz means in Hardy spaces on Dirichlet groupshttps://www.zbmath.org/1483.430062022-05-16T20:40:13.078697Z"Defant, Andreas"https://www.zbmath.org/authors/?q=ai:defant.andreas"Schoolmann, Ingo"https://www.zbmath.org/authors/?q=ai:schoolmann.ingoIn a recent series of papers the authors have developed a theory of Hardy spaces of general Dirichlet series, closely connected with harmonic analysis on groups. Given a frequency \(\lambda = (\lambda_{n})_{n}\) (i.e., strictly increasing and unbounded), they introduced in [\textit{A. Defant} and \textit{I. Schoolmann}, J. Fourier Anal. Appl. 25, No. 6, 3220--3258 (2019; Zbl 1429.43004)] the notion of \(\lambda\)-Dirichlet group (which defines a family of characters \((h_{\lambda_{n}})_{n}\)). For such a group \(G\), they also defined the Hardy space \(H_{p}^{\lambda}(G)\) for \(1 \leq p \leq \infty\). Here they deal with the convergence of the Riesz means for functions in these spaces.
Given \(f \in H_{1}^{\lambda}(G)\), the first \((\lambda,k)\)-Riesz sum of length \(x >0\) is defined as
\[
R^{\lambda,k}_{x}(f) = \sum_{\lambda_{n}<x} \hat{f}(h_{\lambda_{n}}) \Big( 1 - \frac{\lambda_{n}}{x} \Big)^{k} h_{\lambda_{n}} \,.
\]
The main result of the paper shows that, for every \(k>0\), the expression
\[
R^{\lambda,k}_{\max}(f) (\omega) = \sup_{x >0} \big\vert R^{\lambda,k}_{x}(f) (\omega) \big\vert \,,
\]
for \(f \in H_{1}^{\lambda}(G)\) and \(\omega \in G\), defines a bounded sublinear operator
\[
R^{\lambda,k}_{\max} : H_{1}^{\lambda}(G) \to L_{1,\infty}(G)
\]
and
\[
R^{\lambda,k}_{\max} : H_{p}^{\lambda}(G) \to L_{p}(G) \text{ for } 1 < p \leq \infty .
\]
As a consequence, \(R^{\lambda,k}_{x}(f)(\omega)\) converges (in \(x\)) to \(f(\omega)\) for almost every \(\omega\).
When horizontal translations are considered, the situation improves. It is shown that for \(u,k > 0\), there exists a constant \(C=C(u,k)\) so that for every frequency \(\lambda\), all \(1 \leq p \leq \infty\) and \(f \in H_{p}(G)^{\lambda}\) we have
\[
\bigg( \int_{G} \sup_{x >0} \Big\vert \sum_{\lambda_{n} < x} \hat{f} (h_{\lambda_{n}}) e^{-u\lambda_{n}} \Big( 1 - \frac{\lambda_{n}}{x} \Big)^{k} h_{\lambda_{n}} (\omega) \Big\vert^{p} d \omega \bigg)^{1/p} \leq C \Vert f \Vert_{p} \,.
\]
Note that in this case the inequality holds even for \(p=1\), and that the constant does not depend on \(p\).
One of the main tools to prove the main result is a maximal Hardy-Littlewood operator, adapted to this setting. If \((G, \beta)\) is a Dirichlet group and \(f \in L_{1}(G)\), then for almost every \(\omega \in G\) the function defined by \(f_{\omega}(t) = f(\omega \beta(t))\) is locally integrable on \(\mathbb{R}\). It is proved that the adapted Hardy-Littlewood maximal operator, given by
\[
\overline{M}(f) (\omega) = \sup_{\genfrac{}{}{0pt}{2}{I \subset \mathbb{R}}{\text{interval}}} \frac{1}{\vert I \vert} \int_{I} \vert f_{\omega} (t) \vert dt
\]
defines a sublinear bounded operator \(\overline{M}: L_{1}(G) \to L_{1,\infty}(G)\) and \(\overline{M}: L_{p}(G) \to L_{p}(G)\) for \(1 < p \leq \infty\).
It is known that, for \(1 < p < \infty\) and any frequency \(\lambda\), the sequence \((h_{\lambda_{n}})\) is a Schauder basis of \(H_{p}^{\lambda}(G)\) and, therefore the Riesz means of any function \(f\) converge (in norm) to \(f\). Here it is proved that this is also the case for \(p=1\), that is
\[
\lim_{x \to \infty} \big\Vert R^{\lambda, k}_{x}(f) - f \Vert_{1} =0
\]
for every \(k>0\) and every \(f \in H_{1}^{\lambda}(G)\).
Applications of all these are given to general Dirichlet series and to almost periodic functions.
Reviewer: Pablo Sevilla Peris (Valencia)The factorisation property of \(\ell^\infty(X_k)\)https://www.zbmath.org/1483.460142022-05-16T20:40:13.078697Z"Lechner, Richard"https://www.zbmath.org/authors/?q=ai:lechner.richard"Motakis, Pavlos"https://www.zbmath.org/authors/?q=ai:motakis.pavlos"Müller, Paul F. X."https://www.zbmath.org/authors/?q=ai:muller.paul-f-x"Schlumprecht, Thomas"https://www.zbmath.org/authors/?q=ai:schlumprecht.thomasSummary: In this paper we consider the following problem: let \(X_k\) be a Banach space with a normalised basis \((e_{(k, j)})_j\), whose biorthogonals are denoted by \((e_{(k,j)}^*)_j\), for \(k\in\mathbb{N}\), let \(Z=\ell^\infty(X_k:k\in\mathbb{N})\) be their \(\ell^\infty\)-sum, and let \(T:Z\to Z\) be a bounded linear operator with a large diagonal, i.e.,
\[
\inf\limits_{k,j}\left|e^*_{(k,j)}(T(e_{(k,j)})\right|>0.
\]
Under which condition does the identity on \(Z\) factor through \(T\)? The purpose of this paper is to formulate general conditions for which the answer is positive.Invariant spaces of entire functionshttps://www.zbmath.org/1483.460212022-05-16T20:40:13.078697Z"Krivosheev, A. S."https://www.zbmath.org/authors/?q=ai:krivosheev.aleksandr-sergeevich"Krivosheeva, O. A."https://www.zbmath.org/authors/?q=ai:krivosheeva.o-aLet \(D\subset\mathbb{C}\) be a convex domain and let \(H(D)\) be the space of holomorphic functions on \(D\) endowed with the compact open topology. The paper under review deals with the following problem: Let \(W\) be an invariant subspace of the differentiation operator on \(H(D)\). Which conditions ensure that all functions of \(W\) can be extended to entire functions? This problem naturally arises from the problem of expanding convergence domains of exponential series and their special cases, power series and Dirichlet series. \(W\) is assumed to satisfy \textit{spectral synthesis}, i.e., the closure of the span of the eigenvectors of the differentiation operator in \(H(D)\) is the whole \(W\). The following subset of the unit circle \(\mathbb{T}\) is defined,
\[
J(D)= \Bigl\{\omega\in \mathbb T: \ \sup_{z\in D}\text{Re}\,z\omega=+\infty \Bigr\}.
\]
Let \(\Delta:=\{\lambda_k: k\in\mathbb{N}\}\) be the sequence of eigenvalues of the differentiation operator acting on \(W\). Let \(\Xi(\Delta):=\{\overline{\lambda}/|\lambda|: \lambda\in \Delta\} \). The main theorem asserts that the continuation problem has a positive solution when \(\Xi(\Delta)\subset J(D)\). This result was known only under the assumption that \(J(D)\) is open in \(\mathbb{T}\).
Reviewer: Enrique Jordá (Alicante)RKH spaces of Brownian type defined by Cesàro-Hardy operatorshttps://www.zbmath.org/1483.460232022-05-16T20:40:13.078697Z"Galé, José E."https://www.zbmath.org/authors/?q=ai:gale.jose-e"Miana, Pedro J."https://www.zbmath.org/authors/?q=ai:miana.pedro-j"Sánchez-Lajusticia, Luis"https://www.zbmath.org/authors/?q=ai:sanchez-lajusticia.luisSummary: We study reproducing kernel Hilbert spaces introduced as ranges of generalized Cesàro-Hardy operators, in one real variable and in one complex variable. Such spaces can be seen as formed by absolutely continuous functions on the positive half-line (or paths of infinite length) of fractional order, in the real case. A theorem of Paley-Wiener type is given which connects the real setting with the complex one. These spaces are related with fractional operations in the context of integrated Brownian processes. We give estimates of the norms of the corresponding reproducing kernels.On the Fock kernel for the generalized Fock space and generalized hypergeometric serieshttps://www.zbmath.org/1483.460242022-05-16T20:40:13.078697Z"Park, Jong-Do"https://www.zbmath.org/authors/?q=ai:park.jong-doSummary: In this paper, we compute the reproducing kernel \(B_{m, \alpha}(z, w)\) for the generalized Fock space \(F_{m, \alpha}^2(\mathbb{C})\). The usual Fock space is the case when \(m=2\). We express the reproducing kernel in terms of a suitable hypergeometric series \({}_1 F_q\). In particular, we show that there is a close connection between \(B_{4, \alpha}(z, w)\) and the error function. We also obtain the closed forms of \(B_{m, \alpha}(z, w)\) when \(m=1,2/3,1/2\). Finally, we also prove that \(B_{m, \alpha}(z, z)\sim e^{\alpha |z|^m} |z|^{m-2}\) as \(|z|\longrightarrow\infty\).Some essentially normal weighted composition operators on the weighted Bergman spaceshttps://www.zbmath.org/1483.470472022-05-16T20:40:13.078697Z"Fatehi, Mahsa"https://www.zbmath.org/authors/?q=ai:fatehi.mahsa"Shaabani, Mahmood Haji"https://www.zbmath.org/authors/?q=ai:shaabani.mahmood-hajiSummary: First of all, we obtain a necessary and sufficient condition for a certain operator \(T_{w}C_{\varphi}\) to be compact on \(A^{2}_{\alpha}\). Next, we give a short proof for Proposition 2.5 which was proved by \textit{B. D. MacCluer} et al. [Complex Var. Elliptic Equ. 58, No. 1, 35--54 (2013; Zbl 1285.47031)]. Then, we characterize the essentially normal weighted composition operators \(C_{\psi, \varphi}\) on the weighted Bergman spaces \(A^{2}_{\alpha}\), when \(\varphi \in \mathrm{LFT} (\mathbb D)\) is not an automorphism and \(\psi \in H^\infty\) is continuous at a point \(\zeta\) which \(\varphi\) has a finite angular derivative. After that, we find some non-trivially essentially normal weighted composition operators, when \(\varphi \in \mathrm{LFT} (\mathbb D)\) is not an automorphism. In the last section, for \(\varphi \in \mathrm{AUT} (\mathbb D)\) and \(\psi \in {A} (\mathbb D)\), we characterize the essentially normal weighted composition operators \(C_{\psi, \varphi}\) on \(A^{2}_{\alpha}\) and investigate some essentially normal weighted composition operators \(C_{\psi, \varphi}\) on \(H^2\) and \(A^{2}_{\alpha}\). Finally, we find some non-trivially essentially normal weighted composition operators \(C_{\psi, \varphi}\) on \(H^2\) and \(A^{2}_{\alpha}\), when \(\varphi \in \mathrm{AUT} (\mathbb D)\) and \(\psi \in {A} (\mathbb D)\).Differences of generalized weighted composition operators from the Bloch space into Bers-type spaceshttps://www.zbmath.org/1483.470492022-05-16T20:40:13.078697Z"Liu, Xiaosong"https://www.zbmath.org/authors/?q=ai:liu.xiaosong"Li, Songxiao"https://www.zbmath.org/authors/?q=ai:li.songxiaoSummary: We study the boundedness and compactness of the differences of two generalized weighted composition operators acting from the Bloch space to Bers-type spaces.Libera operator on mixed norm spaces \(H_{\nu}^{p,q,\alpha}\) when \(0 < p < 1\)https://www.zbmath.org/1483.470602022-05-16T20:40:13.078697Z"Jevtić, Miroljub"https://www.zbmath.org/authors/?q=ai:jevtic.miroljub"Karapetrović, Boban"https://www.zbmath.org/authors/?q=ai:karapetrovic.bobanSummary: Results from [\textit{M. Pavlović}, ``Definition and properties of the libera operator on mixed norm spaces'', Sci. World J. 2014, Article ID 590656, 15 p. (2014; \url{doi:10.1155/2014/590656})] on Libera operator acting on mixed norm spaces \(H_{\nu}^{p,q,\alpha}\), for \(1 \leq p \leq \infty\), are extended to the case \(0 < p < 1\).\(C\)-selfadjointness of the product of a composition operator and a maximal differentiation operatorhttps://www.zbmath.org/1483.470632022-05-16T20:40:13.078697Z"Shaabani, Mahmood Haji"https://www.zbmath.org/authors/?q=ai:shaabani.mahmood-haji"Fatehi, Mahsa"https://www.zbmath.org/authors/?q=ai:fatehi.mahsa"Hai, Pham Viet"https://www.zbmath.org/authors/?q=ai:pham-viet-hai.Summary: Let \(\varphi\) be an automorphism of \(\mathbb{D}\). In this paper, we consider the operator \(C_\varphi D_{\psi_0,\psi_1}\) on the Hardy space \(H^2\) which is the product of composition and the maximal differential operator. We characterize these operators which are \(C\)-selfadjoint with respect to some conjugations \(C\). Moreover, we find all Hermitian operators \(C_\varphi D_{\psi_0,\psi_1}\), when \(\varphi\) is a rotation.Norm of some operators from logarithmic Bloch-type spaces to weighted-type spaceshttps://www.zbmath.org/1483.470642022-05-16T20:40:13.078697Z"Stević, Stevo"https://www.zbmath.org/authors/?q=ai:stevic.stevoSummary: Operator norm of weighted composition operators from the iterated logarithmic Bloch space \(\mathcal{B}_{{\log}_k} , k\in \mathbb{N}\), or the logarithmic Bloch-type space \(\mathcal{B}_{{\log}^{\beta}},\beta \in (0,1)\) to weighted-type spaces on the unit ball are calculated. It is also calculated norm of the product of differentiation and composition operators among these spaces on the unit disk.Co-spherical electronic configuration of the helium-like atomic systemshttps://www.zbmath.org/1483.811622022-05-16T20:40:13.078697Z"Liverts, Evgeny Z."https://www.zbmath.org/authors/?q=ai:liverts.evgeny-zSummary: The properties of a special configuration of a helium-like atomic system, when both electrons are on the surface of a sphere of radius \(r\), and angle \(\theta\) characterizes their positions on sphere, are investigated. Unlike the previous studies, \(r\) is considered as a quantum mechanical variable but not a parameter. It is important that the ``co-spherical'' and the ``collinear'' configuration are coincident in two points. For \(\theta=0\) one obtains the state of the electron-electron coalescence, whereas the angle \(\theta=\pi\) characterizes the \textbf{e-n-e} configuration when the electrons are located at the ends of the diameter of sphere with the nucleus at its center. The Pekeris-like method representing a fully three-body variational technique is used for the expedient calculations. Some interesting features of the expectation values representing the basic characteristics of the ``co-spherical'' electronic configuration are studied. The unusual properties of the expectation values of the operators associated with the kinetic and potential energy of the two-electron atom/ion possessing the ``co-spherical'' configuration are found. Refined formulas for calculations of the two-electron Fock expansion by the Green's function approach are presented. The model wave functions of high accuracy describing the ``co-spherical'' electronic configuration are obtained. All results are illustrated in tables and figures.