On the structure of Bergman and poly-Bergman spaces
Nikolai Vasilevski1999, Integral Equations and Operator Theory
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
Let II be the upper half-plane in C, consider the Bergman space A2(II), the subspace of all analytic functions from Lz(II). The complete decomposit~on of L2(II) onto Bergman and Bergman type spaces of poly-analytic and poly-antianalytic functions is obtained. The orthogonal Bergman type projections onto each of these subspaces are described. Connections with the Hardy spaces and the Szego projections are established. where Pn-l in the orthogonal projection (4.6) of LI(R+) onto the one-dimensional space Ln-1, generated by the function I,-l(y), defined in (4.2). PROOF. Follows directly from Corollaries 4.2 and 4.4.
Key takeaways
- Consider now the space L2(II) and its Bergman subspace A2(II), consisting of all analytic in II functions.
- We will use the following ortonormal base in Lz(R+): {I,(y)),"=,, where = e~f '~, (~) , L,(y) is the Laguerre polinomial of degree n. Denote by L, the one-dimensional subspace of L2(W+), generated by f,(y), and by P, the orthogonal projection of L2(W+) onto L,.
- Then the image of the Bergman space A: = U1(A2(II)) can be described as the (closed) subspace of L2(II), which consists of all functions satisfying the equation
- where Lo is the one-dimensional subspace of L2(P+), generated by fo = e-#Ia,
- where Lo is the one-dimensional subspace of L2(R+), generated by fo = e-vla,
Related papers
Poly-Bergman projections and orthogonal decompositions of L2-spaces over bounded domains
Operator Theory: Advances and Applications, Vol. 181, 2008
The paper is devoted to obtaining explicit representations of poly-Bergman and anti-poly-Bergman projections in terms of the singular integral operators S D and S * D on the unit disk D, studying relations between different true poly-Bergman and true anti-poly-Bergman spaces on the unit disk that are realized by the operators S D and S * D , establishing two new orthogonal decompositions of the space L 2 (U, dA) (in terms of poly-Bergman and antipoly-Bergman spaces) for an arbitrary bounded open set U ⊂ C with the Lebesgue area measure dA, considering violation of Dzhuraev's formulas and establishing explicit forms of the Bergman and anti-Bergman projections for several open sectors. . Primary 32A25, 46E22; Secondary 30G30, 31A10, 47G10.
Two Elementary Analytic Functions and Their Relationship with Hardy and Bergman Spaces
2018
In this paper, we start by proving that the function which is holomorphic in the open unit disc centred at the origin, is an element of a Hardy space if and only if Here we give a new proof for a known result. Moreover, the present work provides two different new proofs for one of the implications mentioned above. One proves that the same function is an element of a Bergman space if and only if This is the first completely new result of this work. From these theorems we deduce the behavior of the function in the half – open disc Although the assertions claimed above refer to complex analytic functions, and the involved spaces are function spaces of analytic complex functions, the proofs from below are based on results and methods of real analysis.
Duality Properties of Non-Reflexive Bergman Space of The Upper Half-Plane
2019
The study of duality properties of the spaces of analytic functions continues to attract the attention of many mathematicians. Most studies have concentrated on the reflexive Hardy and Bergman spaces both on the unit disk and the upper half-plane. For instance, Zhu, Peloso, among others have determined the duality properties of Hardy and Bergman spaces. For the non-reflexive Bergman spaces of the disk, it was proved by Axler that the dual and the predual are identified as big and little Bloch spaces respectively. For non-reflexive Bergman spaces of the upper half-plane La(U, μα), the dual is well known as the Bloch space B∞(U, i) but the predual is not known. In our study therefore, we have determined the predual of La(U, μα). We have also determined the group of weighted composition operators defined on predual space of La(U, μα) and investigated both its semigroup and spectral properties. To determine the predual space of La(U, μα), we used the Cayley transform as well as related ...
A note on weighted poly-Bergman spaces
2020
We introduce a la Vasilevski the weighted poly-Bergman spaces in the unit disc and provide concrete orthonormal basis and give close expression of their reproducing kernel. The main tool in the description if these spaces is the so-called disc polynomials that form an orthogonal basis of the whole weighted Hilbert space.
On Generators of the Hardy and the Bergman Spaces
arXiv (Cornell University), 2023
A function ϕ which is analytic and bounded in the unit disk D is called a generator for the Hardy space H 2 (D) or the Bergman space A 2 (D) if polynomials in ϕ are dense in the corresponding space. We characterize generators in terms of ϕ−invariant subspaces which are also z−invariant and study wandering properties of such subspaces. Density of bounded analytic functions in the ϕ−invariant subspaces of H 2 (D) is also investigated.
Projections for harmonic Bergman spaces and applications
Journal of Functional Analysis, 2004
On the setting of general bounded smooth domains in R n ; we construct L 1 -bounded nonorthogonal projections and obtain related reproducing formulas for the harmonic Bergman spaces. In addition, we show that those projections satisfy Sobolev L p -estimates of any order even for p ¼ 1: Among applications are Gleason's problems for the harmonic Bergman-Sobolev and (little) Bloch functions on star-shaped domains with strong reference points. r 2004 Elsevier Inc. All rights reserved. MSC: primary 31B05; secondary 31B10
A space of projections on the Bergman space
We define a set of projections on the Bergman space A 2 parameterized by an affine closed space of a Banach space. This family is defined from an affine space of a Banach space of holomorphic functions in the disk and includes the classical Forelli-Rudin projections.
Polyharmonic Bergman spaces and Bargmann type transforms
Journal of Mathematical Analysis and Applications, 2017
In the main result of the paper we prove the decomposition of polyharmonic Bergman spaces over the upper-half plane into spaces of polyanalytic functions. Then, we introduce the decomposition of polyharmonic Bergman spaces into the orthogonal sum of its true polyharmonic Bergman subspaces and we state isometric isomorphisms between the different true polyharmonic Bergman spaces. This allows us to define the k-th harmonic Hilbert component of a polyharmonic Bergman function and to prove closed formulas for the reproducing kernel functions of the true polyharmonic and the polyharmonic Bergman spaces. The harmonic complex Fourier transform is introduced in order to give an explicit description of the cartesian and the Laguerre harmonic components of the images of a Bargmann type transform for the true polyharmonic Bergman spaces. Finally, it is proved that the polyharmonic Bergman space of order j is isometric isomorphic to 2j copies of the corresponding Hardy space.
Polyharmonic Bergman Spaces on Half-Spaces and Bargmann Type Transforms
Several properties of the polyharmonic Bergman space over the upper-half plane are described partially based on some special properties of the two-sided compression of the Beurling-Ahlfors transform. We introduce the geometrically clear decomposition of polyharmonic Bergman spaces into the orthogonal sum of its true polyharmonic Bergman subspaces and we state isometric isomorphisms between the different true polyharmonic Bergman spaces. This allows us to define the k-th harmonic Hilbert component of a function u in the polyharmonic Bergman space. We then prove closed formulas for the reproducing kernel functions of the true poly-harmonic and the polyharmonic Bergman spaces. The harmonic complex Fourier transform is introduced in order to give an explicit description of the cartesian and the Laguerre harmonic components of the images of a Bargmann type transform for the true polyharmonic Bergman spaces. Finally, it is proved that the polyharmonic Bergman space of order j is isometric...
New Characterizations of Bergman Spaces
Ann. Acad. Sci. Fenn. Math, 2008
We obtain several new characterizations for the standard weighted Bergman spaces A p α on the unit ball of C n in terms of the radial derivative, the holomorphic gradient, and the invariant gradient.
The Bergman projection in spaces of entire functions
Annales Polonici Mathematici, 2012
We establish L p-estimates for the weighted Bergman projection on a nonsingular cone. We apply these results to the weighted Fock space with respect to the minimal norm in C n .
A CHARACTERISATION OF BERGMAN SPACES ON THE UNIT BALL OF ℂn
Glasgow Mathematical Journal, 2009
It was shown in [2] that a holomorphic function f in the unit ball B n of C n belongs to the weighted Bergman space A p α , p > n + 1 + α, if and only if the function |f
Bergman Projection and Bergman Spaces
Journal of Operator …, 2001
Abstract. In this paper we study mapping properties of the Bergman pro-jection P, ie which function spaces or classes are preserved by P. It is shown that the Bergman projection is of weak type (1, 1) and bounded on the Orlicz space Lϕ(D, dA) iff Lϕ(D, dA) is reflexive. So the dual ...
Toeplitz operators in polyanalytic Bergman type spaces
Functional Analysis and Geometry, 2019
We consider Toeplitz operators in Bergman and Fock type spaces of polyanalytic $L^2\textup{-}$functions on the disk or on the half-plane with respect to the Lebesgue measure (resp., on $\mathbb{C}$ with the plane Gaussian measure). The structure involving creation and annihilation operators, similar to the classical one present for the Landau Hamiltonian, enables us to reduce Toeplitz operators in true polyanalytic spaces to the ones in the usual Bergman type spaces, however with distributional symbols. This reduction leads to describing a number of properties of the operators in the title, which may differ from the properties of the usual Bergman-Toeplitz operators.
Recent progress and open problems in the Bergman space
The following text is a modified and updated version of the problem collection , which was written in 1993 but became publicly available only in 1995. It was a survey of various open problems; a general survey of the field was provided in [41, 42] in 1998, written in 1995 and 1996, respectively. Since then, a number of new developments have taken place, which in their turn have led to new questions. We feel it is time to update the problem collection.
Bergman and Bloch spaces of vector-valued functions
Mathematische Nachrichten, 2003
Multiplication operators on the Bergman space via analytic continuation
2011
In this paper, we develop a machinery to study multiplication operators on the Bergman space via the Hardy space of the bidisk. We show that only a multiplication operator by a finite Blaschke product has a unique reducing subspace on which its restriction is unitarily equivalent to the Bergman shift. Using the machinery we study the structure of reducing subspaces unitary equivalence of a multiplication operator on the Bergman space. As a consequence, we completely classify reducing subspaces of the multiplication operator by a Blaschke product φ with order three on the Bergman space to solve a conjecture of Zhu and obtain that the number of minimal reducing subspaces of the multiplication operator equals the number of connected components of the Riemann surface of φ −1 • φ over D.
Hyponormality on general Bergman spaces
Filomat, 2019
A bounded operator T on a Hilbert space is hyponormal if T*T-TT* is positive. We give a necessary condition for the hyponormality of Toeplitz operators on weighted Bergman spaces, for a certain class of radial weights, when the symbol is of the form f+g?, where both functions are analytic and bounded on the unit disk. We give a sufficient condition when f is a monomial.
Holomorphic N K and Bergman-type spaces
Birkh¨auser Verlarg Publisher Basel/Switzerland, 2008
In this paper we introduce a new class of functions, called NK-type space of analytic functions by the help of a nondecreasing function K : [0, ∞) → [0, ∞). Further, under mild conditions on the weight function K we characterize lacunary series in NK space. Finally, we study the boundedness and compactness of composition operators between NK and Bergman spaces. Mathematics Subject Classification (2000). Primary 47B33; 47B38 Secondary 30H05.
Related topics
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.