Weighted Bergman Spaces Research Articles

COMPACT AND HILBERT–SCHMIDT WEIGHTED COMPOSITION OPERATORS ON WEIGHTED BERGMAN SPACES

AbstractLet u and $\varphi $ be two analytic functions on the unit disk D such that $\varphi (D) \subset D$ . A weighted composition operator $uC_{\varphi }$ induced by u and $\varphi $ is defined on $A^2_{\alpha }$ , the weighted Bergman space of D, by $uC_{\varphi }f := u \cdot f \circ \varphi $ for every $f \in A^2_{\alpha }$ . We obtain sufficient conditions for the compactness of $uC_{\varphi }$ in terms of function-theoretic properties of u and $\varphi $ . We also characterize when $uC_{\varphi }$ on $A^2_{\alpha }$ is Hilbert–Schmidt. In particular, the characterization is independent of $\alpha $ when $\varphi $ is an automorphism of D. Furthermore, we investigate the Hilbert–Schmidt difference of two weighted composition operators on $A^2_{\alpha }$ .

Ces\xe0ro-like operators acting on spaces of analytic functions

Let {mathbb {D}} be the unit disc in {mathbb {C}}. If mu is a finite positive Borel measure on the interval [0, 1) and f is an analytic function in {mathbb {D}}, f(z)=sum _{n=0}^infty a_nz^n (zin {mathbb {D}}), we define Cμ(f)(z)=∑n=0∞μn∑k=0nakzn,z∈D,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\mathcal {C}}_\\mu (f)(z)= \\sum _{n=0}^\\infty \\mu _n\\left( \\sum _{k=0}^na_k\\right) z^n,\\quad z\\in {\\mathbb {D}}, \\end{aligned}$$\\end{document}where, for nge 0, mu _n denotes the n-th moment of the measure mu , that is, mu _n=int _{[0, 1)}t^ndmu (t). In this way, {mathcal {C}}_mu becomes a linear operator defined on the space {mathrm{Hol}}({mathbb {D}}) of all analytic functions in {mathbb {D}}. We study the action of the operators {mathcal {C}}_mu on distinct spaces of analytic functions in {mathbb {D}}, such as the Hardy spaces H^p, the weighted Bergman spaces A^p_alpha , BMOA, and the Bloch space {mathcal {B}}.

Extremal problems for vanishing functions in Bergman spaces

We prove two sharp estimates for the subspace of a standard weighted Bergman space that consists of functions vanishing at a given point (with prescribed multiplicity).

Weighted composition operators on Bergman spaces Aωp$A^p_\omega$

AbstractLet ϕ be an analytic self‐map of the open unit disk , and let u be an analytic function on . The weighted composition operator induced by ϕ with weight u is given by for z in and f analytic on . In this paper, we study weighted composition operators acting between two exponentially weighted Bergman spaces and . We characterize the bounded, compact and Schatten class membership operators acting from to when and . To obtain this, we first get an important estimate for the norm of the reproducing kernel in and some new characterizations of Carleson measures. Our results use certain integral transforms that generalize the usual Berezin transform. In the case where and , we compare our criteria with those given by Kriete and MacCluer in [15].

Complex symmetric weighted composition-differentiation operators

In this article, we mainly study the weighted composition-differentiation operator on the weighted Bergman space and the derivative Hardy space , which characterize complex symmetric weighted composition-differentiation operator . Moreover, the normality and self-adjointness of are also discussed.

Reverse Carleson inequalities for weighted Bergman spaces with Békollé weights

Reverse Carleson inequalities for weighted Bergman spaces with Békollé weights

Toeplitz operators on Bergman spaces with exponential weights

In this paper, we focus on the weighted Bergman spaces in with . We first give characterizations of those finite positive Borel measures µ in such that the embedding is bounded or compact for . Then we describe bounded or compact Toeplitz operators from one Bergman space to another for all possible . Finally, we characterize Schatten class Toeplitz operators on .

Product-type operators on Banach spaces of analytic functions on the upper half-plane

Let , the upper half-plane in the complex plane . Let be the set of all analytic functions on Π and be the set of all analytic self-maps on Π. For , and , where , , define an operator In this paper, we characterize the boundedness and compactness of the operator on the Hardy, weighted-Bergman spaces on Π to weighted-type spaces on Π.

Carleson measures on the generalized Hartogs triangles

In this paper, we characterize the Carleson measures of the weighted Bergman spaces on a wide class of non-smooth pseudoconvex Reinhardt domains, which contains some generalizations of the classical Hartogs triangle. In addition, we obtain a necessary condition for a finite positive Borel measure μ to be an vanishing Carleson measure of the weighted Bergman space on the generalized Hartogs triangles.

Toeplitz Operators on Weighted Bergman Spaces Induced by a Class of Radial Weights

Suppose that \(\omega \) is a radial weight on the unit disk that satisfies both forward and reverse doubling conditions. Using Carleson measures and T1-type conditions, we obtain necessary and sufficient conditions of the positive Borel measure \(\mu \) such that the Toeplitz operator \(T_{\mu ,\omega }:L^p_a(\omega )\rightarrow L_a^1(\omega )\) is bounded and compact for \(0<p\le 1\). In addition, we obtain a bump condition for the bounded Toeplitz operators with \(L^1(\omega )\) symbol on \(L^1_a(\omega )\). This generalizes a result of Zhu in (J Funct Anal 87(1):31-50, 1989).

The $\partial$-operator and real holomorphic vector fields

Let $(M,h)$ be a Hermitian manifold and $\psi$ a smooth weight function on $M$. The $\partial$-complex on weighted Bergman spaces $A^2_{(p,0)}(M,h, e^{-\psi})$ of holomorphic $(p,0)$-forms was recently studied in [[10] and [9]. It was shown that if $h$ is K\"ahler and a suitable density condition holds, the $\partial$-complex exhibits an interesting holomorphicity/duality property when $(\bar\partial\psi)^{\sharp}$ is holomorphic (i.e., when the real gradient field $\mathrm{grad}_h\psi$ is a real holomorphic vector field). For general Hermitian metrics this property does not hold without the holomorphicity of the torsion tensor $T_p{}^{rs}$. In this paper, we investigate the existence of real-valued weight functions with real holomorphic gradient fields on K\"ahler and conformally K\"ahler manifolds and their relationship to the $\partial$-complex on weighted Bergman spaces. For K\"ahler metrics with multi-radial potential functions on $\mathbb C^n$ we determine all multi-radial weight functions with real holomorphic gradient fields. For conformally K\"ahler metrics on complex space forms we first identify the metrics having holomorphic torsion leading to several interesting examples such as the Hopf manifold $\mathbb{S}^{2n-1} \times \mathbb{S}^1$, and the "half" hyperbolic metric on the unit ball. For some of these metrics, we further determine weight functions $\psi$ with real holomorphic gradient fields. They provide a wealth of triples $(M,h,e^{-\psi})$ of Hermitian non-K\"ahler manifolds with weights for which the $\partial$-complex exhibits the aforementioned holomorphicity/duality property. Among these examples, we study in detail the $\partial$-complex on the unit ball with the half hyperbolic metric and derive a new estimate for the $\partial$-equation.

Complex symmetric weighted composition-differentiation operators of order $n$ on the weighted Bergman spaces

We study the complex symmetric structure of weighted composition--differentiation operators of order $n $ on the weighted Bergman spaces $A_{\alpha}^2$ with respect to some conjugations. Then we provide some examples of these operators.

Essential Norms of Stević–Sharma Operators from General Banach Spaces into Zygmund‐Type Spaces

A Stević–Sharma operator denoted by is a generalization product of multiplication, differentiation, and composition operators. Using several restrictive terms, we characterize an approximation of the essential norm of the Stević–Sharma operator from a general class X of holomorphic function spaces into Zygmund‐type spaces with some of the most convenient test functions on the open unit disk. As an application, we show that our results hold up for several other domain spaces of , such as the Hardy space and the weighted Bergman space.

Weighted composition operators and differences of composition operators between weighted Bergman spaces on the ball

In this paper, we estimate essential norms of weighted composition operators and differences of two composition operators on the weighted Bergman spaces in the unit ball.

N-Best Kernel Approximation in Reproducing Kernel Hilbert Spaces

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 have 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.

Volterra integral operator from weighted Bergman spaces to general function spaces

Volterra integral operator from weighted Bergman spaces to general function spaces

Mean-squared approximation of some classes of complex variable functions by Fourier series in the weighted Bergman space 𝐵2,𝛾

Mean-squared approximation of some classes of complex variable functions by Fourier series in the weighted Bergman space 𝐵2,𝛾

$$\mathbb{K}$$-homogeneous tuple of operators on bounded symmetric domains

Let Ω be an irreducible bounded symmetric domain of rank r in ℂd. Let $$\mathbb{K}$$ be the maximal compact subgroup of the identity component G of the biholomorphic automorphism group of the domain Ω. The group $$\mathbb{K}$$ consisting of linear transformations acts naturally on any d-tuple T = (T1, …, Td) of commuting bounded linear operators. If the orbit of this action modulo unitary equivalence is a singleton, then we say that T is $$\mathbb{K}$$ -homogeneous. In this paper, we obtain a model for a certain class of $$\mathbb{K}$$ -homogeneous d-tuple T as the operators of multiplication by the coordinate functions z1, …, zd on a reproducing kernel Hilbert space of holomorphic functions defined on Ω. Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class, (iii) unitary equivalence and similarity of these d-tuples. In particular, we show that the adjoint of the d-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class B1(Ω). For an irreducible bounded symmetric domain Ω of rank 2, an explicit description of the operator $$\sum\nolimits_{i = 1}^d {T_i^\ast {T_i}} $$ is given. In general, based on this formula, we make a conjecture giving the form of this operator.

Isometries of the product of composition operators on weighted Bergman space

In this paper, the necessary and sufficient conditions for the product of composition operators to be isometry are obtained on weighted Bergman space. With the help of a counter example we also proved that unlike on [Formula: see text] and [Formula: see text] the composition operator on [Formula: see text] induced by an analytic self-map on [Formula: see text] with fixed origin need not be of norm one. We have generalized the Schwartz’s [Composition operators on [Formula: see text], thesis, University of Toledo (1969)] well-known result on [Formula: see text] which characterizes the almost multiplicative operator on [Formula: see text]

Contractive inequalities for mixed norm spaces and the Beta function

For a wide range of pairs of mixed norm spaces such that one space is contained in another, we characterize all cases when contractive norm inequalities hold. In particular, this yields such results for many pairs of weighted Bergman spaces. Some inequalities of this type are motivated by their applications in Number Theory and in Mathematical Physics.