Search

Let ρ be the defining function of a bounded strongly pseudoconvex domain D with smooth boundary in C n . In this paper, we study the essential norm of Hankel operators H f β which are considered as operators from weighted Bergman spaces A p (D , | ρ | α d V) to L q (D , | ρ | β d V) with 1 < p ≤ q < ∞ and - 1 < α , β < ∞ . For f ∈ L 1 (D , | ρ | β d V) , we obtain some quantities in terms of the symbol function f, which are comparable to the essential norm of the Hankel operator H f β . Furthermore, it is shown that the essential norm of H f β is equivalent to the distance norm from itself to compact Hankel operators. [ABSTRACT FROM AUTHOR]

We study Laplace equation and heat equation with the Gaussian white noise as the boundary/initial condition. Using the stochastic pre-orthogonal adaptive Fourier decomposition on the Bergman space, we give sparse expansions of the solutions of the boundary/initial value problems. The proposed methodology breaks away from the H - H K structure. The key point to the solution is to use the Wiener isometry property of the Gaussian white noise integral and the corresponding Green's function to represent the covariance of the solution. For the heat equation with the Gaussian white noise initial value, we define a class of parabolic Bergman space following E. Stein, prove existence of Bergman kernel, and obtain its expression, and further verify the boundary vanishing property needed for generating a sparse expansion. In order to make the heat flow satisfy a long-time decay property, we induce a subspace of the parabolic Bergman space, which corresponds to a class of homogeneous Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper, we develop the theory of weighted Bergman space and obtain a general representation formula of the Bergman kernel function for the spaces on the Reinhardt domain containing the origin. As applications, we calculate the concrete forms of the Bergman kernels for some special weights on the Reinhardt domains $ \mathbb {C}^n $ C n , $ D_{n,m}:= \{(z, w)\in \mathbb {C}^n \times \mathbb {C}^m : \|w\|^2 D n , m := { (z , w) ∈ C n × C m : ‖ w ‖ 2 < e − μ 1 ‖ z ‖ μ 2 } and $ V_{\eta }:=\{\left (z, z', w\right) \in \mathbb {C}^{n} \times \mathbb {C}^{m} \times \mathbb {C} : \sum _{j=1}^{n} e^{\eta _{j}|w|^{2}}\left |z_{j}\right |^{2}+\|z'\|^{2} V η := { (z , z ′ , w) ∈ C n × C m × C : ∑ j = 1 n e η j | w | 2 | z j | 2 + ‖ z ′ ‖ 2 < 1 }. [ABSTRACT FROM AUTHOR]

We study Laplace equation and heat equation with the Gaussian white noise as the boundary/initial condition. Using the stochastic pre-orthogonal adaptive Fourier decomposition on the Bergman space, we give sparse expansions of the solutions of the boundary/initial value problems. The proposed methodology breaks away from the H - H K structure. The key point to the solution is to use the Wiener isometry property of the Gaussian white noise integral and the corresponding Green's function to represent the covariance of the solution. For the heat equation with the Gaussian white noise initial value, we define a class of parabolic Bergman space following E. Stein, prove existence of Bergman kernel, and obtain its expression, and further verify the boundary vanishing property needed for generating a sparse expansion. In order to make the heat flow satisfy a long-time decay property, we induce a subspace of the parabolic Bergman space, which corresponds to a class of homogeneous Sobolev spaces. [ABSTRACT FROM AUTHOR]

Let ω be a doubling weight and 0 < p ≤ q < ∞ . The essential norm of Riemann–Stieltjes operator T g from the weighted Bergman space A ω p to A ω q was investigated in the unit ball of ℂ n . [ABSTRACT FROM AUTHOR]

We study the problem of recovering a signal from magnitudes of its wavelet frame coefficients when the analyzing wavelet is real-valued. We show that every real-valued signal can be uniquely recovered, up to global sign, from its multiwavelet frame coefficients \[ \{\lvert \mathcal {W}_{\phi _i} f(\alpha ^{m}\beta n,\alpha ^{m}) \rvert : i\in \{1,2,3\}, m,n\in \mathbb {Z}\} \] for every \alpha >1,\beta >0 with \beta \ln (\alpha)\leq 4\pi /(1+4p), p>0, when the three wavelets \phi _i are suitable linear combinations of the Poisson wavelet P_p of order p and its Hilbert transform \mathscr {H}P_p. For complex-valued signals we find that this is not possible for any choice of the parameters \alpha >1,\beta >0, and for any window. In contrast to the existing literature on wavelet sign retrieval, our uniqueness results do not require any bandlimiting constraints or other a priori knowledge on the real-valued signals to guarantee their unique recovery from the absolute values of their wavelet coefficients. [ABSTRACT FROM AUTHOR]

Given a doubling weight ω on the unit disk D , let Aωp be the space of all the holomorphic functions f, where ∥ f ∥ A p := (∫ D ∣ f (z) ∣ p (z) d A (z)) 1 / p < ∞. We completely characterize the topological connectedness of the set of composition operators on Aωp. As an application, we construct an interesting example which reveals that two composition operators on Aαp in the same path component may fail to have a compact difference and give a negative answer to the Shapiro-Sundberg question in the (standard) weighted Bergman space. In addition, we completely describe the central compactness of any finite linear combinations of composition operators on Aωp in three terms: a Julia-Carathéodory-type function-theoretic characterization, a power-type characterization, and a Carleson-type measure-theoretic characterization. [ABSTRACT FROM AUTHOR]

Herein, the Laplace transform representations for functions of weighted holomorphic Bergman spaces on the tube domains are developed. Then a weighted version of the edge-of-the-wedge theorem is derived as a byproduct of the main results. [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]

We consider the weighted Bergman space A α 2 (픻) of analytic functions f on the unit disc 픻 = {z ∈ ℂ :|z| < 1} such that f A α 2 2 := ∫ D f z 2 d A α z < + ∞ , where d Aα(z) = (α+1) (1 −|z|2)α d A(z) and d A z = d x d y π (the normalized Lebesgue area measure). We investigate the generalized Riccati operator equation X A 1 X + A 2 X A 3 + A 4 Y A 5 + A 6 = 0 with Ai ∈ 퓑( A α 2 (픻)), i = 1, 6. Also we study the solvability of the operator equation X 1 T φ 1 + X 2 T φ 2 + ⋯ + X n T φ n = I A α 2 , in the set of Toeplitz operators on the weighted Bergman space A α 2 (픻). Moreover, we characterize compactness of the operator Tφ Tψ − Tη in terms of Berezin transforms. [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]

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]

We obtain a new boundedness criterion for the difference of two composition operators from a weighted Bergman space A α p into a Lebesgue space L q (μ) , where 0 < q < p and α > - 1 . As a consequence, we provide a direct proof that such a bounded difference operator is necessarily compact. We also characterize compact differences of composition operators from A α p into A β q explicitly for 0 < p ≤ q and α , β > - 1 . [ABSTRACT FROM AUTHOR]

This article studies the boundedness of the inclusion mapping from weighted Bergman spaces Aαp into a class of tent type space Tsp,n(μ). As an application, the boundedness, compactness and essential norm of generalized integral operators Tgn,k and Sgn,0 from Aαp to general function spaces are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, we discuss block Toeplitz operators T Φ whose symbols are in the class of normal functions Φ (z) = G ∗ (z) + F (z) , where F (z) = A m z m + A N z N and G (z) = A - m z m + A - N z N (A i ∈ M 2) , on the weighted Bergman space. Additionally, we present a necessary and sufficient condition for the hyponormality of the block Toeplitz operators T Φ . [ABSTRACT FROM AUTHOR]

Let Mu be the multiplication operator with symbol u which is a holomorphic function on the open unit ball B in Cn, Cφ the composition operator with symbol φ which is a holomorphic self-map of B, and Rm, m ∈ N,the mth iterated radial derivative operator. The metrical boundedness and metrical compactness of the operator CφRmMu from weighted Bergman spaces to weighted-type spaces on B are characterized. As an application of the proofs, the same properties of the operator ... from weighted Bergman spaces to weighted-type spaces on B are characterized. [ABSTRACT FROM AUTHOR]

We study composition operators between the weighted Bergman space of the polydisk and the weighted Bergman space of the unit disk. We find conditions on the symbols to guarantee the boundedness and compactness of the composition operator. We shall also find a necessary and sufficient condition for the composition operator to be Hilbert–Schmidt. [ABSTRACT FROM AUTHOR]

We study the dominating sets and reverse Carleson measures on exponentially weighted Bergman spaces A ω p under a new metric. Then, we give some applications of reverse Carleson measure and a generalization of Theorem 1 in Luecking [Sampling measures for Bergman spaces on the unit disk. Math Ann. 2000;316:659–679]. [ABSTRACT FROM AUTHOR]

We introduce a polynomial differentiation composition operator on spaces of holomorphic functions on the open unit ball in the n-dimensional complex vector space, and characterize the boundedness and compactness of the operator from the classical weighted Bergman space to the weighted-type space and the little weighted-type space of holomorphic functions on the unit ball. [ABSTRACT FROM AUTHOR]

In this article, we mainly study the weighted composition-differentiation operator on the weighted Bergman space A α 2 and the derivative Hardy space S 1 2 , which characterize complex symmetric weighted composition-differentiation operator D u , φ . Moreover, the normality and self-adjointness of D u , φ are also discussed. [ABSTRACT FROM AUTHOR]

It is known that the norm of the Hilbert matrix operator on weighted Bergman spaces A α p was conjectured by Karapetrović to be π sin ⁡ (α + 2) π p when α > − 1 and p > α + 2. The conjecture has been confirmed by Božin and Karapetrović in the case α = 0. In this paper we prove the conjecture for the cases both α = 1 and 0 < α ≤ 1 47. Moreover, we also show that the conjecture is valid when − 1 < α < 0 and p ≥ 2 (α + 2). [ABSTRACT FROM AUTHOR]

In this paper, we consider the hyponormality of Toeplitz operators acting on the weighted Bergman space A α 2 (D). We establish necessary or sufficient conditions for the hyponormality of Toeplitz operators with non-harmonic symbols φ (z) that are bivariate polynomials in z and z ¯ (i.e., φ (z) = a z m z ¯ n + b z s z ¯ t ). In particular, we present some characteristics according to the coefficients and degree of φ (z). [ABSTRACT FROM AUTHOR]

We characterize wandering subspace property of the shift operator on zero-based invariant subspaces of reproducing kernel Hilbert spaces by using the kernels of Toeplitz operators. As applications, we prove that the wandering subspace property of the shift operator holds on the weighted Bergman space A α 2 ( α > 1) for zero-based invariant subspaces when the zeros lie in a small disk centred at the origin. [ABSTRACT FROM AUTHOR]

Motivated by the recent work of Galindo et al. (J. Funct. Anal. 265, 629–643, 2013), in this paper, we give an elegant compactness criterion for any finite linear combination of composition operators on the weighted Bergman space in terms of power type characterization. More precisely, let T be any finite linear combination of composition operators, then T is compact on A α p (D) if and only if lim n → ∞ ∥ T z n ∥ A − γ ∥ z n ∥ A − γ = 0 , which reveals that the compactness of T on the weighted Bergman space A α p (D) and Korenblum space A−γ(D) are equivalent. [ABSTRACT FROM AUTHOR]

We obtain the norm and essential norm estimates for the difference of two weighted composition operators from a weighted Bergman space A α p into a Lebesgue space L q (μ) , where 0 < p ≤ q , α > - 1 and μ is a positive Borel measure on the unit disk of C . The weights of the weighted maps in this paper are not necessarily analytic or bounded. Consequently, characterizations for the boundedness and compactness of the difference operator are established. [ABSTRACT FROM AUTHOR]

Let φ and u be analytic maps on the open unit disc D such that φ (D) ⊂ D and H (D) be the spaces of analytic functions on D. For a non-negative integer n, the generalized weighted composition operator Dφ,un is defined by Dφ,un f = u. (f(n) o φ), f ∈ H (D). In this paper, we characterize the boundedness of the generalized weighted composition operator Dφ,un between the weighted Bergman spaces. We also compute the upper and lower bounds for the essential norm of this operator. [ABSTRACT FROM AUTHOR]

We give a concrete description of complex symmetric monomial Toeplitz operators T z p z ¯ q on the weighted Bergman space A2(Ω), where Ω denotes the unit ball or the unit polydisk. We provide a necessary condition for T z p z ¯ q to be complex symmetric. When p,q ∈ ℕ2, we prove that T z p z ¯ q is complex symmetric on A2(Ω) if and only if p1 = q2 and P2 = q1. Moreover, we completely characterize when monomial Toeplitz operators T z p z ¯ q on A 2 (D n) are JU-symmetric with the n × n symmetric unitary matrix U. [ABSTRACT FROM AUTHOR]

Holomorphic functions on the maximal Grauert tube of a hyperbolic compact Riemann surface are studied. It is shown that their weighted Bergman spaces are infinite dimensional for arbitrary weight order greater than - 1 in spite of the fact that they do not admit any non-constant bounded holomorphic functions. The key ingredient of the proof is a computation of weighted norms of analytic continuations of eigenfunctions of the Laplacian in terms of the hypergeometric function. This result complements our previous work (Adachi in Trans Am Math Soc 374(10):7499–7524, 2021) where it was shown that the space of geodesic segments on the Riemann surface has exactly the same property. [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]

We obtain conditions that imply reverse Carleson inequalities for weighted Bergman spaces with Békollé weights. Our main theorem generalizes the results in the unweighted case by Luecking. [ABSTRACT FROM AUTHOR]

The spectrum of a weighted composition operator C ψ , φ that is induced by an automorphism has been investigated for over 50 years. However, many results are got only under the condition that the weight function ψ is continuous up to the boundary. In this paper we study the spectra and essential spectra of C ψ , φ on weighted Bergman spaces when φ is an elliptic automorphism, without the assumption that ψ is continuous up to the boundary. [ABSTRACT FROM AUTHOR]

We completely describe the boundedness and compactness of Hankel operators with general symbols acting on Bergman spaces with exponential type weights. [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]

For weighted Bergman spaces on the unit disk, we give trace formulas of semicommutators of Toeplitz operators with C 2 (D ‾) symbols. We generalize this formula to weighted Bergman spaces on the unit ball in higher dimensions. Applications and examples on the Hankel operators are also discussed. [ABSTRACT FROM AUTHOR]

We study the Helton-Howe trace and the Connes-Chern character for Toeplitz operators on weighted Bergman spaces via the idea of quantization. We prove a local formula for the large t -limit of the Connes-Chern character as the weight goes to infinity. And we show that the Helton-Howe trace of Toeplitz operators is independent of the weight t and obtain a local formula for the Helton-Howe trace for all weighted Bergman spaces using harmonic analysis and quantization. [ABSTRACT FROM AUTHOR]

By making a seminal use of the maximum modulus principle of holomorphic functions we prove existence of n -best kernel approximation for a wide class of reproducing kernel Hilbert spaces of holomorphic functions in the unit disc, and for the corresponding class of Bochner type spaces of stochastic processes. This study thus generalizes the classical result of n -best rational approximation for the Hardy space and a recent result of n -best kernel approximation for the weighted Bergman spaces of the unit disc. The type of approximations has significant applications to signal and image processing and system identification, as well as to numerical solutions of the classical and the stochastic type integral and differential equations. [ABSTRACT FROM AUTHOR]

Let G be a finite pseudoreflection group and Ω ⊆ C d be a bounded domain which is a G -space. We establish identities involving Toeplitz operators on the weighted Bergman spaces of Ω and Ω / G using invariant theory and representation theory of G. This, in turn, provides techniques to study algebraic properties of Toeplitz operators on the weighted Bergman space on Ω / G. We specialize on the generalized zero-product problem and characterization of commuting pairs of Toeplitz operators. As a consequence, more intricate results on Toeplitz operators on the weighted Bergman spaces on some specific quotient domains (namely symmetrized polydisc, monomial polyhedron, Rudin's domain) have been obtained. [ABSTRACT FROM AUTHOR]