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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]

The main goal of this paper was to completely characterize complex symmetric difference of the weighted composition operators induced by three type symbols on weighted Bergman space of the right half-plane with the conjugations J f (z) = f (z ¯) ¯ , J s f (z) = f (z ¯ + i s) ¯ , and J ∗ f (z) = 1 z α + 2 f (1 z ¯ ) ¯ . The special phenomenon that we focus on is that the difference is complex symmetric on weighted Bergman spaces of the half-plane with the related conjugation if and only if each weighted composition operator is complex symmetric. The main goal of this paper was to completely characterize complex symmetric difference of the weighted composition operators induced by three type symbols on weighted Bergman space of the right half-plane with the conjugations , , and . The special phenomenon that we focus on is that the difference is complex symmetric on weighted Bergman spaces of the half-plane with the related conjugation if and only if each weighted composition operator is complex symmetric. [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 investigate some properties of the harmonic Bergman spaces A p (σ) on a q-homogeneous tree, where q ≥ 2 , 1 ≤ p < ∞ , and σ is a finite measure on the tree with radial decreasing density, hence nondoubling. These spaces were introduced by J. Cohen, F. Colonna, M. Picardello and D. Singman. When p = 2 they are reproducing kernel Hilbert spaces and we compute explicitely their reproducing kernel. We then study the boundedness properties of the Bergman projector on L p (σ) for 1 < p < ∞ and their weak type (1,1) boundedness for radially exponentially decreasing measures on the tree. The weak type (1,1) boundedness is a consequence of the fact that the Bergman kernel satisfies an appropriate integral Hörmander's condition. [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]

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]

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 completely characterize the boundedness and compactness of the differences for composition operators on the weighted Bergman spaces with Békollé weights in the open unit ball and estimate the norm and essential norm of weighted composition operators. [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]

In this paper we establish uncertainty inequality of Heisenberg type for Hardy space H, Dirichlet space D and Bergman space B, respectively. Next, we introduce a weighted Hardy space Hβ. This space which gives a generalization of some Hilbert spaces of analytic functions on the unit disk like, the Hardy space H, the Dirichlet space D and the Bergman space B, it plays a background to our contribution. Especially, we study the derivative operator D and its adjoint operator Lβ on Hβ, and we deduce a general uncertainty inequality of Heisenberg type for this space. [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]

In this paper, we consider the H-Toeplitz and Toeplitz operators acting on the Bergman space. First, we describe the characterizations of commutativity of two H-Toeplitz operators with certain harmonic symbols. For the general case, it seems very hard. As an extension to the study of Toeplitz operators on the Bergman space, we present the necessary and suffcient conditions of the commutativity of the H-Toeplitz operator and the Toeplitz operator with non-harmonic symbols. [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]

In this paper, the order boundedness and essential norm of generalized weighted composition operators on Bergman spaces with doubling weights are characterized. Specially, we estimate the essential norm of these operators on weighted Bergman spaces by using the reduce order method. [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]

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]

One of the aims of the present paper is to obtain some properties about logarithmic Bergman-type space on the unit ball. As some applications, the bounded and compact operators Sm ~u;' = Pmi =0 MuiC'<i from logarithmic Bergman-type space to weighted-type space on the unit ball are completely characterized. [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]

In this paper, we characterize when the product of two H-Toeplitz operators to be another H-Toeplitz with one general and another quasihomogeneous symbols. Also, we describe the product of H-Toeplitz operator and Toeplitz operator to be another H-Toeplitz with certain harmonic symbols. [ABSTRACT FROM AUTHOR]

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]

For a fixed analytic function g on the unit disc, we consider the analytic paraproducts induced by g, which are formally defined by T g f (z) = ∫ 0 z f (ζ) g ′ (ζ) d ζ , S g f (z) = ∫ 0 z f ′ (ζ) g (ζ) d ζ , and M g f (z) = g (z) f (z). We are concerned with the study of the boundedness of operators in the algebra A g generated by the above operators acting on Hardy, or standard weighted Bergman spaces on the disc. The general question is certainly very challenging, since operators in A g are finite linear combinations of finite products (words) of T g , S g , M g which may involve a large amount of cancellations to be understood. The results in [1] show that boundedness of operators in a fairly large subclass of A g can be characterized by one of the conditions g ∈ H ∞ , or g n belongs to B M O A or the Bloch space, for some integer n > 0. However, it is also proved that there are many operators, even single words in A g whose boundedness cannot be described in terms of these conditions. The present paper provides a considerable progress in this direction. Our main result provides a complete quantitative characterization of the boundedness of an arbitrary word in A g in terms of a "fractional power" of the symbol g , that only depends on the number of appearances of each of the letters T g , S g , M g in the given word. [ABSTRACT FROM AUTHOR]

In this paper, we describe the minimal reducing subspaces of Toeplitz operators induced by non-analytic monomials on the weighted Bergman spaces and Dirichlet spaces over the unit ball B 2 . It is proved that each minimal reducing subspace M is finite dimensional, and if dim M ≥ 3, then M is induced by a monomial. Furthermore, the structure of commutant algebra ν (T w ¯ N z N ) := { M w N ∗ M z N , M z N ∗ M w N } ′ is determined by N and the two dimensional minimal reducing subspaces of T w ¯ N z N . We also give some interesting examples. [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]

Let R A be the Cartan classical domains of type Ⅲ and Ⅳ, and Δ g is assumed to be the Laplace-Beltrami operator associated to the Bergman metric g on R A . In this paper, we derive an estimate for λ 1 (Δ g) , which is the bottom of the spectrum of Δ g on R A . [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]

Let Ω be a bounded domain in Rn with C0;1 boundary, and let dΩ: Ω ! R be the distance function dΩ (x):= dist (x; Ω): Our aim in this paper is to study the existence and properties of principal eigenvalues of selfadjoint elliptic operators with weight function and singular potential, whose model problem is u + bu = mu in Ω, u = 0 on Ω; u > 0 in Ω where b Ω R is a nonnegative function such that d2Ω b 2 L1 () ; m: Ω R is a nonidentically zero function in L1 () that may change sign, and the solutions are understood in weak sense. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the q-Carleson measure for Dirichlet type spaces D p a on the unit ball when 0 < p < q < ¥ and a > 1 by using Carleson blocks and Carleson tubes. [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]

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]

The aim of the paper is to obtain a interesting result about the atomic decomposition of Bloch-type space in the polydisk. The existing similar results have been applied many times to the atomic decompositions of Bloch-type and weighted Bergman spaces in the unit ball. [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]

The aim of the paper is to obtain a interesting result about the atomic decomposition of Bloch-type space in the polydisk. The existing similar results have been applied many times to the atomic decompositions of Bloch-type and weighted Bergman spaces in the unit ball. [ABSTRACT FROM AUTHOR]

In this paper, we consider little Hankel operators Γφ defined on the Bergman space L²a(D) with symbol φ ∈ H∞(D) that are contractions. Necessary and sufficient conditions are obtained for the existence of a nontrivial unitary part of these little Hankel operators. We also present an explicit description of this unitary part. This extends the results of Butz for Hankel operators defined on the Hardy space. [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]

This paper introduces and studies composition operators on the Bergman space with quasiconformal symbols. We first characterize the boundedness of the norms. Then we discuss operator-theoretic issues including compactness characterization, essential norms, Schatten class membership, and conditions for closed ranges. [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 study the behavior of growth of the modulus of an arbitrary algebraic polynomial in the weighted Bergman space Ap(G, h), p > 0, in exterior points and in closure of region having exterior nonzero and interior zero angles at a finite number of boundary points. Bernstein-Walsh-type estimates are obtained for algebraic polynomials, in bounded regions with a piecewise-smooth boundary. Using an inequality of the Markov-Nikolskii type, the behavior of polynomials in the whole complex plane is given. [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 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 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]