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In the paper 'Composition operators on weighted Bergman spaces of Dirichlet series. J Math Anal Appl. 2015;426:340–363', Bailleul completely characterized the boundedness of composition operators on weighted Bergman spaces of Dirichlet series for the case of symbols with $ c_0\ge 1 $ c 0 ≥ 1. But the sufficient conditions for the other case $ c_0=0 $ c 0 = 0 were unsolved. In this paper, we follow this line and study the boundedness of composition operators on weighted Bergman spaces of Dirichlet series for the case $ c_0=0 $ c 0 = 0. Moreover, we also obtain the compact characterizations of composition operators with $ c_0\geq 1 $ c 0 ≥ 1. [ABSTRACT FROM AUTHOR]

One of the aims of this paper is to characterize 3-complex symmetric weighted composition operators induced by three types of symbols on the weighted Bergman space of the right half-plane with the conjugation J f (z) = f (z ¯) ¯ . It is well known that the complex symmetry is equivalent to 2-complex symmetry for the weighted composition operators studied in the paper. However, the interesting fact that 3-complex symmetry is not equivalent to 2-complex symmetry for such operators is found in the paper. Finally, the complex normal of such operators on the weighted Bergman space of the right half-plane with the conjugation J is characterized. [ABSTRACT FROM AUTHOR]

Suppose $ \alpha >-1 $ α > − 1 and $ 1\leq p \leq \infty $ 1 ≤ p ≤ ∞. Let $ f=\mathcal {P}_{\alpha }[F] $ f = P α [ F ] be an α-harmonic mapping on $ {\mathbb D} $ D with the boundary F being absolute continuous and $ \dot {F}\in L^p(0,2\pi) $ F ˙ ∈ L p (0 , 2 π) , where $ \dot {F}({\rm e}^{i\theta }):=\frac {{\rm d}}{{\rm d}\theta }F({\rm e}^{i\theta }) $ F ˙ (e iθ) := d d θ F (e iθ). In this paper, we investigate the membership of $ f_z $ f z and $ f_{\bar {z}} $ f z ¯ in the space $ \mathcal {H}_{\mathcal {G}}^{p}(\mathbb {D}) $ H G p (D) , the generalized Hardy space. We prove, if $ \alpha >0 $ α > 0 , then both $ f_z $ f z and $ f_{\bar {z}} $ f z ¯ are in $ \mathcal {H}_{\mathcal {G}}^{p}(\mathbb {D}) $ H G p (D). If $ \alpha α < 0 , then $ f_z $ f z and $ f_{\bar {z}}\in \mathcal {H}_{\mathcal {G}}^{p}(\mathbb {D}) $ f z ¯ ∈ H G p (D) if and only if f is analytic. Finally, we investigate a Schwarz Lemma for α-harmonic functions. [ABSTRACT FROM AUTHOR]

In this paper we discuss some basic properties of octonionic Bergman and Hardy spaces. In the first part we review some fundamental concepts of the general theory of octonionic Hardy and Bergman spaces together with related reproducing kernel functions in the monogenic setting. We explain how some of the fundamental problems in well-defining a reproducing kernel can be overcome in the non-associative setting by looking at the real part of an appropriately defined para-linear octonion-valued inner product. The presence of a weight factor of norm 1 in the definition of the inner product is an intrinsic new ingredient in the octonionic setting. Then we look at the slice monogenic octonionic setting using the classical complex book structure. We present explicit formulas for the slice monogenic reproducing kernels for the unit ball, the right octonionic half-space and strip domains bounded in the real direction. In the setting of the unit ball we present an explicit sequential characterization which can be obtained by applying the special Taylor series representation of the slice monogenic setting together with particular octonionic calculation rules that reflect the property of octonionic para-linearity. [ABSTRACT FROM AUTHOR]

In this paper, we first completely characterize the complex symmetric Toeplitz operators T φ on the Hardy spaces H 2 (D) with conjugations C p i , j and C n . Next, we give a method to determine the coefficients of φ (z) when T φ is complex symmetric on H 2 (D) with the conjugation C σ , which partially solves a problem raised by [2]. Finally, we consider the complex symmetric Toeplitz operators T φ on the weighted Bergman spaces A 2 (B n) and the pluriharmonic Bergman spaces b 2 (B n) with conjugations C V , where V is a symmetric permutation matrix. [ABSTRACT FROM AUTHOR]

In this paper, we prove embedding theorems for the Möbius invariant space \mathcal {Q}_p on the open unit ball of \mathbb {C}^n into logarithmic tent spaces in the Bergman metric. [ABSTRACT FROM AUTHOR]

In this paper, we give a universal description of the boundedness and compactness of Toeplitz operator T μ ω between Bergman spaces A η p and A ν q when μ is a positive Borel measure, 1 < p,q < ∞ and ω, η, ν are regular weights. By using Khinchin's inequality and Kahane's inequality, we get a new characterization of the Carleson measure for Bergman spaces induced by regular weights. [ABSTRACT FROM AUTHOR]

Let μ be a self-similar measure with compact support K. The Hausdorff dimension of K is α. The Cauchy transform of μ is denoted by F (z) . For 0 < β < 1 , we define the function F [ β ] , which compares with the fractional derivative of F of order β. Let Φ (z) = F (1 / z) , | z | < 1 . In this paper, we prove that Φ [ β ] belongs to A p for 0 < p < 1 / (β + 1) , and (Φ ′) [ β ] belongs to A p for 1 ≤ p < 1 / β ≤ 1 / (2 − α) , where A p is the Bergman space. At the same time, we give a value distribution property of F, which is similar to the big Picard theorem. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak-type regularity of the Bergman projection on n-dimensional classical Hartogs triangles. We extend the results of Huo–Wick on the 2-dimensional classical Hartogs triangle to the n-dimensional classical Hartogs triangle and show that the Bergman projection is of weak type at the upper endpoint of L q -boundedness but not of weak type at the lower endpoint of L q -boundedness. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of an operator \mathcal T naturally associated to the so-called Collatz map , which maps an integer n \geq 0 to n / 2 if n is even and 3n + 1 if n is odd. This operator \mathcal T is defined on certain weighted Bergman spaces \mathcal B ^2 _\omega of analytic functions on the unit disk. Building on previous work of Neklyudov, we show that \mathcal T is hypercyclic on \mathcal B ^2 _\omega, independently of whether the Collatz Conjecture holds true or not. Under some assumptions on the weight \omega, we show that \mathcal T is actually ergodic with respect to a Gaussian measure with full support, and thus frequently hypercyclic and chaotic. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to study new features and specific properties of the Toeplitz operator with radial symbols in harmonic Fock spaces. A new spectral decomposition of a Toeplitz operator with Wick symbols is also established. [ABSTRACT FROM AUTHOR]

It is well-known that the research of linear combination of composition operators has become a topic of increasing interest. Recently, Choe, Koo and Wang proved that the compactness of combinations composition operators induced by the symbols satisfying the condition (CNC) implies that each difference is compact on the weighted Bergman space. Motivated by that work, in this paper, we discuss which difference is compact on the weighted Bergman space when the coefficients do not satisfy the condition (CNC). [ABSTRACT FROM AUTHOR]

The first goal of this paper is to find a representation of the Fock space on C n in terms of the weighted Bergman spaces of the projective spaces CP n − 1 ; i.e., every function in the Fock space can be written as a direct sum of elements in weighted Bergman space on CP n − 1 . Also, we study the C * algebras generated by Toeplitz operators where the symbols are taken from the following two families of functions: Firstly, the symbols depend on the moment map associated with the unit circle, and secondly, the symbols are invariant under the same action. Moreover, we analyze the commutative relations between these algebras, and we apply these results to find new commutative Banach algebras generated by Toeplitz operators on Fock space of C n . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the relationship between sampling measure \mu, Berezin transform \tilde {\mu } and r-averaging transform \widehat {\mu }_r on Bergman spaces. Compared with some results of Luecking [Amer. J. Math. 107 (1985), pp. 85–111], our results provide an equivalent description of sampling measures, which reveals the reason why 1/\hat {\mu }_r\in L^\infty or 1/\tilde {\mu }\in L^\infty does not make sure that \mu is a sampling measure on Bergman spaces. [ABSTRACT FROM AUTHOR]

Given a structural set ψ ⊂ H , some papers [M. Shapiro, V. Vasilevski, (Complex Var Theory Appl. 1995;27:17–46 and 1995;27:67–96)] study the quaternionic ψ-Fueter operator calculus, where the Fueter-type operator is ψ D = ∑ i = 0 3 ψ i ∂ ∂ x i . The aim of this paper is to present the function theory and the Bergman theory induced by the ψ-Fueter-type operator: α ψ D [ f ] = α f + ψ D [ f ] , where a is a quaternionic constant and f is a continuously differentiable function. Particularly, this work shows the Stokes and Borel–Pompieu formulas associated to α ψ D and presents several results of the Bergman theory such as the existence of the Bergman kernel, Bergman projection and their conformal covariant properties. [ABSTRACT FROM AUTHOR]

In this paper, we study Toeplitz and Hankel operators on exponential weighted Bergman spaces. For 0 < p < ∞ , we obtain sufficient and necessary conditions for Toeplitz and Hankel operators to belong to Schatten-p class by the averaging functions of symbols. For a continuous increasing convex function h, the Schatten-h class Toeplitz and Hankel operators are also characterized. [ABSTRACT FROM AUTHOR]

In this paper, we give a characterization of Fredholmness of the Toeplitz operators on the Bergman spaces A φ p with exponential weights in D when 0 < p < ∞. Also, we obtain the sufficient and necessary conditions which the Toeplitz and Hankel operators on A φ 2 belong to h -Schatten class, where h is a continuous increasing convex function. [ABSTRACT FROM AUTHOR]

We can realise the representations of the group SL2(R) on the unit disc. This is due to the isomorphism between the group SL2(R) and the group SU(1, 1). The discrete series representations for the group SL2(R) given by πn(g)φ(z) = φ (dz − b) / (a − cz) (a − cz) −n, n ∈ Z. (1) are on the Bergman space where n ≥ 2 [5, 6, 10]. Lang [13, IX] studied the discrete series on the group SL(R) in the upper half-plane and on the unit disc. For n = 1, the SL2(R) representation is called the mock discrete series. The representation space of the mock discrete series is the Hardy space [5, 6, 10]. In this paper we describe the SL2(R) representation on the Dirichlet space. [ABSTRACT FROM AUTHOR]

We study extremal problems related to the best joint polynomial approximation of functions analytic in the unit disk and belonging to the Bergman space . The problem of joint approximation of periodic functions and their derivatives by trigonometric polynomials was considered by Garkavy [1] in 1960. Then, in the same year, Timan [2] considered this problem for classes of entire functions defined on the entire axis. The problem of joint approximation of functions and their derivatives is considered in more detail in Malozemov's monograph [3], where some classical theorems of the theory of approximation of functions are presented and generalized. In the present paper, a number of exact theorems are obtained and sharp upper bounds for the best joint approximations of a function and its successive derivatives by polynomials and their respective derivatives on some classes of complex functions belonging to the Bergman space are calculated. [ABSTRACT FROM AUTHOR]

We characterize Bergman-Orlicz spaces of eigenfunctions on hyperbolic space in terms of Lipschitz type conditions with respect to pseudo-hyperbolic, hyperbolic and Euclidean metrics. As an application, some properties of symmetric lifting operators defined by the difference quotients of eigenfunctions are obatined. Our results of this paper are extensions of the corresponding known ones in classical weighted Bergman spaces. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to show that over a large class of bounded domains Ω⊂C², for 1

In 1934, G. H. Hardy and J. E. Littlewood calculated [Proc. London Math. Soc. (2) 36 (1934), pp. 516–531] the optimal Cesàro exponent for Hardy spaces. In this paper we calculate it for mixed norm spaces, hence including the Bergman spaces in particular. The main technical challenge lies in the analysis of the example needed for the critical case. [ABSTRACT FROM AUTHOR]

In this paper, we consider Lu Qi-Keng problem on the Thullen domain. By constructing an invariant, we find a self-reciprocal polynomial which depends only on the dimension and is independent of the parameter p of the Thullen domain, and then get a fixed number po such that the Bergman kernel function has zero points for 0 < p < po. Meanwhile, when the Bergman kernel function has zeros, the topology of the zero variety and the set of boundary points which are the accumulation points of zeros are characterized. [ABSTRACT FROM AUTHOR]

In this paper we introduce a weighted Hardy space ℋ β . This space generalizes some complex Hilbert spaces like the Dirichlet space 풟 , the Bergman space 풜 and the Segal–Bargmann space ℱ . It plays the role of background for our contribution. In particular, we study the derivative operator D and its adjoint operator L β on ℋ β . Furthermore, we establish a general uncertainty inequality of Heisenberg type for the space ℋ β . [ABSTRACT FROM AUTHOR]

In this paper, we focus on the weighted Bergman spaces A φ p in D with φ ∈ W 0 . We first give characterizations of those finite positive Borel measures µ in D such that the embedding A φ p ⊂ L μ q is bounded or compact for 0 < p , q < ∞. Then we describe bounded or compact Toeplitz operators T μ from one Bergman space A φ p to another A φ q for all possible 0 < p , q < ∞. Finally, we characterize Schatten class Toeplitz operators on A φ 2 . [ABSTRACT FROM AUTHOR]

We continue the study of the so‐called Hausdorff–Zhu operators. In this paper, we study the behavior of the singular values of such operators. We prove the general fact that the sequence of singular numbers tends to zero, as well as we prove power‐type estimates for such behavior under additional conditions on the kernel of the operator. We give application of these results to the boundedness of Hausdorff–Zhu operators in general classes of analytic functions in the unit disc and also in some special classes of analytic functions defined in terms of conditions on Taylor or Fourier coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the hyponormality of dual Toeplitz operators on the orthogonal complement of the harmonic Bergman space. First we show that the dual Toeplitz operator with the bounded symbol is hyponormal if and only if it is normal. Then we obtain a necessary and sufficient condition for the dual Toeplitz operator Sφ with the symbol φ (z) = a z n 1 − m 1 z + b z n 2 − m 2 z (n1, n2, m1, m 2 ∈ N and a, b ∈ ℂ) to be hyponormal. Finally, we show that the rank of the commutator of two dual Toeplitz operators must be an even number if the commutator has a finite rank. [ABSTRACT FROM AUTHOR]

In this paper, we show that the Bergman functions on the Siegel upper half-space enjoy the following uniqueness property: if f\in A_t^p(\mathcal {U}) and \mathcal {L}^{\alpha } f\equiv 0 for some nonnegative multi-index \alpha, then f\equiv 0, where \mathcal {L}^{\alpha }≔(\mathcal {L}_1)^{\alpha _1} \cdots (\mathcal {L}_n)^{\alpha _n} with \mathcal {L}_j = \frac {\partial }{\partial z_j} + 2i \bar {z}_j \frac {\partial }{\partial z_n} for j=1,\ldots, n-1 and \mathcal {L}_n = \frac {\partial }{\partial z_n}. As a consequence, we obtain a new integral representation for the Bergman functions on the Siegel upper half-space. In the end, as an application, we derive a result that relates the Bergman norm to a "derivative norm", which suggests an alternative definition of the Bloch space and a notion of the Besov spaces over the Siegel upper half-space. [ABSTRACT FROM AUTHOR]

In this paper, we study some properties of the nth-order weighted reduced Bergman kernels for planar domains, n ≥ 1. Specifically, we look at Ramadanov type theorems, localization, and boundary behavior of the weighted reduced Bergman kernel and its higher-order counterparts. We also give a transformation formula for these kernels under biholomorphisms. [ABSTRACT FROM AUTHOR]

In this paper we study Toeplitz and Cesàro-type operators on holomorphic function spaces on a homogeneous Siegel domain of Type II. We prove several necessary conditions and sufficient conditions for these operators to be continuous or compact, or to belong to suitable Schatten classes. [ABSTRACT FROM AUTHOR]

In this paper, we consider the d-boundedness of the Bergman metric and a vanishing theorem of L2-cohomology on a class of Hartogs domain, whose base domain is the production of two irreducible bounded symmetric domains of the first type, by using the Bergman kernel function, invariant function, holomorphic automorphism group and so on. [ABSTRACT FROM AUTHOR]

In this paper, we study the complex symmetry of Toeplitz operators on the weighted Bergman spaces over the unit polydisk. First, we completely characterize when anti-linear weighted composition operators W ψ , φ J are conjugations. We then give a sufficient and necessary condition for Toeplitz operators to be complex symmetric with respect to these conjugations. As a consequence, some interesting higher-dimensional complex symmetric phenomena appear on the unit polydisk such as the monomial Toeplitz operators T z p z ¯ q with p = V q for some symmetric permutation matrix V. Surprisingly, these operators T z V q z ¯ q are the only ones that are complex symmetric monomial Toeplitz operators on the unit bidisk. [ABSTRACT FROM AUTHOR]

In this paper, we study some properties of weighted composition operators on a class of weighted Bergman spaces A φ p with 0 < p ≤ ∞ and φ ∈ W 0 . Also, we completely characterize the q-Carleson measure for A φ p in terms of the averaging function and the generalized Berezin transform with 0 < q < ∞. As applications, the boundedness and compactness of weighted composition operators acting from one Bergman space A φ p to another A φ q are equivalently described and the Schatten class property of the weighted composition operator acting on A φ 2 are given. Our main results are expressed in terms of certain Berezin type integral transforms. [ABSTRACT FROM AUTHOR]

Let be a subdomain of C and let be a positive Borel measure on. In this paper, we study the asymptotic behavior of the eigenvalues of compact Toeplitz operators acting on Bergman spaces on. Let. T be the decreasing sequence of the eigenvalues of T, and let be an increasing function such that .n/=nA is decreasing for some A > 0. We give an explicit necessary and sufficient geometric condition on in order to have As applications, we consider composition operators C', acting on some standard analytic spaces on the unit disc D. First, we give a general criterion ensuring that the singular values of C' satisfy sn. Next, we focus our attention on composition operators with univalent symbols, where we express our general criterion in terms of the harmonic measure of '.. We finally study the case where @'. meets the unit circle in one point and give several concrete examples. Our method is based on upper and lower estimates of the trace of h. where h is a suitable concave or convex function. [ABSTRACT FROM AUTHOR]

In this paper, we characterize that the boundedness, compactness and spectral structure of dual Toeplitz operators acting on the orthogonal complement of the harmonic Bergman space. This generalizes the corresponding results for dual Toeplitz operators on the orthogonal complement of the Bergman space due to Stroethoff and Zheng's paper [Trans. Amer. Math. Soc., 354, 2495–2520 (2002)]. [ABSTRACT FROM AUTHOR]

For 0 < p < ∞ , we completely characterize the Schatten p -class embedding and fractional embedding of the Bergman-type space A α 2 on the unit ball of C m. The main results of this paper are twofold: (1) For α , β ∈ R , we show that the necessary and sufficient condition for the embedding operator I : A α 2 → A β 2 is in the Schatten p -class is that p (β − α) − 2 m > 0. (2) For a positive Borel measure μ on B m and a real number τ , we prove that the fractional embedding operator I α + τ τ : A α 2 → L 2 (B m , d μ) is in the Schatten p -class if and only if the Berezin transform of μ , denoted by μ ˜ α α + τ , is in L p 2 (B m , d λ) , where d λ (z) = (1 − | z | 2) − m − 1 d v (z) is the Möbius invariant volume measure and d v is the Lebesgue measure on B m , or equivalently, the averaging function of μ , denoted by μ ˆ r α , is in L p 2 (B m , d λ). • For p > 0 , we completely characterize the Schatten p -class embedding and fractional embedding of the Bergman-type space on the unit ball. • The range 0 < p < 2 is an open problem left by H. T. Kaptanoğlu (see [5]). • We prove that the fractional embedding operator is in Schatten p -class iff the Berezin transform of the positive Borel measure is in L − space. [ABSTRACT FROM AUTHOR]

Let |$f(z)=\sum _{n=0}^{\infty }a_n z^n$| be a formal power series with complex coefficients. Let |$({\mathcal{R}} f)(z)= \sum _{n=0}^{\infty }\pm a_n z^n$| be the randomization of |$f$| by choosing independently a random sign for each coefficient. Let |$H^p({\mathbb{D}})$| and |$L^p_a({\mathbb{D}})$|   |$(p>0)$| denote the Hardy space and the Bergman space, respectively, over the unit disk in the complex plane. In 1930, Littlewood proved that if |$f \in H^2({\mathbb{D}})$|⁠ , then |${\mathcal{R}} f \in H^p({\mathbb{D}})$| for any |$p \in (0, \infty)$| almost surely, and if |$f \notin H^2({\mathbb{D}})$|⁠ , then |${\mathcal{R}} f \notin H^p({\mathbb{D}})$| for any |$p \in (0, \infty)$| almost surely. In this paper, we obtain a characterization of the pairs |$(p, q) \in (0, \infty)^2$| such that |${\mathcal{R}} f$| is almost surely in |$L^q_a({\mathbb{D}})$| whenever |$f \in L^p_a({\mathbb{D}})$|⁠ , including counterexamples to show the optimality of the embedding. In contrast to Littlewood's theorem, random Bergman functions exhibit no improvement of regularity for any |$p>0$|⁠ , but the loss of regularity for |$p<2$| is not as drastic as the Hardy case; there is indeed a nontrivial boundary curve given by |$\frac{1}{q}-\frac{2}{p}+\frac{1}{2}=0$|⁠. Several other results about random Bergman functions are established along the way. The technical difficulties, especially when |$p<1$|⁠ , are different from the Hardy space and we devise a different route of proof. The Dirichlet space follows as a corollary. An improvement of the original Littlewood theorem is obtained. [ABSTRACT FROM AUTHOR]

A Stevi c ′ -Sharma operator denoted by T ψ 1 , ψ 2 , φ is a generalization product of multiplication, differentiation, and composition operators. In this paper, we characterize the bounded and compact Stevi c ′ -Sharma operator T ψ 1 , ψ 2 , φ from a general class X of Banach function spaces into Zygmund-type spaces with some of the most convenient test functions on the open unit disk. Using several restrictive terms, we show that all bounded operators T ψ 1 , ψ 2 , φ from X into the little Zygmund-type spaces are compact. As an application, we show that our results hold up for some other domain spaces of T ψ 1 , ψ 2 , φ , such as the Hardy space and the weighted Bergman space. [ABSTRACT FROM AUTHOR]

In this paper, we study weighted composition operators and differences of composition operators. We use the Carleson measure to estimate the operator norm and the essential norm of those operators which act between two Bergman spaces with Békollé weights. [ABSTRACT FROM AUTHOR]

The generalized derivative Hardy space S α , β 2 (D) consists of all functions whose derivatives are in the Hardy and Bergman spaces as follows: for positive integers α, β, S α , β 2 (D) = { f ∈ H (D) : ∥ f ∥ S α , β 2 2 = ∥ f ∥ H 2 2 + α + β α β ∥ f ′ ∥ A 2 2 + 1 α β ∥ f ′ ∥ H 2 2 < ∞ } , where H (D) denotes the space of all functions analytic on the open unit disk D . In this paper, we study characterizations for Toeplitz operators to be complex symmetric on the generalized derivative Hardy space S α , β 2 (D) with respect to some conjugations C ξ , C μ , λ . Moreover, for any conjugation C, we consider the necessary and sufficient conditions for complex symmetric Toeplitz operators with the symbol φ of the form φ (z) = ∑ n = 1 ∞ φ ˆ (− n) ‾ z ‾ n + ∑ n = 0 ∞ φ ˆ (n) z n . Next, we also study complex symmetric Toeplitz operators with non-harmonic symbols on the generalized derivative Hardy space S α , β 2 (D) . [ABSTRACT FROM AUTHOR]

In this paper, we give a characterization of a complex symmetric Toeplitz operator T φ on the weighted Bergman space A α 2 (D). We first give properties of complex symmetric Toeplitz operators T φ on A α 2 (D). Next, we prove that if T φ is complex symmetric with finite symbol, then T φ is hyponormal on A α 2 (D) if and only if it is hyponormal on the Hardy space H 2 (T). Finally, we consider the complex symmetric Toeplitz operator T φ on A α 2 (D) when the conjugation is a special case. [ABSTRACT FROM AUTHOR]

In Koo and Wang (Joint Carleson measure and the difference of composition operators on A α p (B n). J Math Anal Appl. 2014;419:1119–1142), the authors introduced a concept of joint Carleson measure and used it to characterize when the difference of two composition operators on weighted Bergman space over the unit ball is bounded or compact. In this paper, we extend the concept of joint Carleson measure to the polydisk setting and obtain analogue characterizations of the boundedness (compactness, resp.) of the difference of two composition operators on the weighted Bergman spaces over the unit polydisk, which may provide a unified approach for various ad hoc studies on the boundedness or the compactness of the difference of composition operators on polydisk. Moreover, we construct a concrete example to show that both the boundedness and the compactness depend on the index p when the dimension n ≥ 2 , which is in sharp contrast with the one-variable case where the boundedness and the compactness of the difference of two composition operators are independent of p>0. Due to the complexity of the Carleson measure on the unit polydisk, some new techniques are required in the polydisk setting. [ABSTRACT FROM AUTHOR]

The paper is devoted to weighted estimates for operators of fractional integration of variable order of Bergman type in generalized variable Hölder spaces of holomorphic functions on the unit disc D . Due to the choice of the weight, we can include in consideration the case when the real part of the complex power of the operator degenerates. We prove the estimates of Zygmund type for the modulus of continuity, and then we obtain the corresponding weighted boundedness result. [ABSTRACT FROM AUTHOR]

Let ϕ be an analytic self‐map of the open unit disk D$\mathbb {D}$, and let u be an analytic function on D$\mathbb {D}$. The weighted composition operator induced by ϕ with weight u is given by (uCϕf)(z)=u(z)f(ϕ(z))$(uC_{\phi }f)(z)=u(z)f(\phi (z))$ for z in D$\mathbb {D}$ and f analytic on D$\mathbb {D}$. In this paper, we study weighted composition operators acting between two exponentially weighted Bergman spaces Aωp$A^p_{\omega }$ and Aωq$A^q_{\omega }$. We characterize the bounded, compact and Schatten class membership operators uCϕ$ uC_{\phi }$ acting from Aωp$A^p_{\omega }$ to Aωq$A^q_{\omega }$ when 0

In the paper we develop the Gelfand theory for a new class of commutative Banach algebras generated by Toeplitz operators both in the Bergman and the Hardy space cases. We introduce the so-called generalized pseudo-homogeneous symbols, the functions of the form c (θ , t 1 t ¯ 2) , where (t 1 , t 2) ∈ T × T , thus t 1 t ¯ 2 ∈ T , and c ∈ L ∞ ([ 0 , π 2 ] × T). This class of symbols is a result of step-by-step generalization of previously considered (and studied) symbols. For the commutative algebras under study we describe the compact sets of their maximal ideals, Gelfand transform, and give a criterion on whether our algebra is semi-simple or has a radical. In the last case, we give a complete characterization of the radical. [ABSTRACT FROM AUTHOR]

In this paper, for 1 ≤ p < ∞ , we provide several descriptions of Schatten p -class Hankel operators H f and H f ‾ on the weighted Bergman space A ω 2 , in terms of a certain global and local mean oscillation of the symbol f ∈ L ω 2 , provided ω is in a class of regular weights. The approaches applied to rely on several classical methods, and simultaneously rely on a novel but more convenient construction associated with the atomic decomposition of A ω 2. [ABSTRACT FROM AUTHOR]

In this paper, the boundedness and compactness of the difference of composition-differentiation operators D φ − D ψ acting from Hardy spaces H p to weighted Bergman spaces A α q are completely characterized for all 0 < p , q < ∞. [ABSTRACT FROM AUTHOR]

In this paper, we focus on Carleson embeddings from Bergman spaces A p into L p (μ) , where μ is a positive measure on the unit disk. We describe when this injection is r -summing on A p. We complete the full characterization of such operators when p > 1 , and r ≥ 1. As an immediate application, we get the characterization of absolutely summing weighted composition operators on Bergman spaces. In passing we also prove a new connection between the boundedness of the Berezin transform and the Carleson embedding on Bergman spaces. [ABSTRACT FROM AUTHOR]

In this paper we have shown that if φ ∈ h∞(D) and T(α)φ is the Toeplitz operator with symbol φ defined on the weighted Bergman space L²a(dAα) and if the set {(T(α)φ)*T(α)φ f,(T(α)φ)* f, T(α)f,f} fois linearly dependent for all f ∈ L2a(dAα) then either φ is a constant function or there exists λα, µα ∈ C such that φ-µα/λα is a real-valued function in h∞(D). Here h∞(D) is the set of all bounded harmonic functions on the open unit disk D. [ABSTRACT FROM AUTHOR]

In this paper, we present necessary conditions for the hyponormality of Toeplitz operator Tϕ on the Bergman space La²(D) when the symbol ϕ is a polynomial in z and z̄. [ABSTRACT FROM AUTHOR]