NORM OF THE HILBERT MATRIX OPERATOR ON THE WEIGHTED BERGMAN SPACES

2017 ◽  
Vol 60 (3) ◽  
pp. 513-525 ◽  
Author(s):  
BOBAN KARAPETROVIĆ
Keyword(s):  
Lower Bound ◽  
Bergman Space ◽  
Upper Bound ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Matrix Operator ◽  
Weighted Bergman Spaces ◽  
Formula Group ◽  
Hilbert Matrix ◽  
Image Position

AbstractWe find the lower bound for the norm of the Hilbert matrix operator H on the weighted Bergman space Ap,α \begin{equation*} \|H\|_{A^{p,\alpha}\rightarrow A^{p,\alpha}}\geq\frac{\pi}{\sin{\frac{(\alpha+2)\pi}{p}}}, \,\, \textnormal{for} \,\, 1<\alpha+2<p. \end{equation*} We show that if 4 ≤ 2(α + 2) ≤ p, then ∥H∥Ap,α → Ap,α = $\frac{\pi}{\sin{\frac{(\alpha+2)\pi}{p}}}$, while if 2 ≤ α +2 < p < 2(α+2), upper bound for the norm ∥H∥Ap,α → Ap,α, better then known, is obtained.

Isometries of Weighted Bergman Spaces

1982 ◽  
Vol 34 (4) ◽  
pp. 910-915 ◽  
Author(s):  
Clinton J. Kolaski
Keyword(s):  
Bergman Space ◽  
Bergman Spaces ◽  
Holomorphic Functions ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Bounded Symmetric Domains ◽  
General Argument ◽  
Image Position ◽  
Linear Isometries

In [2], [8] and [10], Forelli, Rudin and Schneider described the isometries of the Hp spaces over balls and polydiscs. Koranyi and Vagi [6] noted that their methods could be used to describe the isometries of the Hp spaces over bounded symmetric domains. Recently Kolaski [4] observed that the algebraic techniques used above and Rudin's theorem on equimeasurability extended to the Bergman spaces over bounded Runge domains. In this paper we use the same general argument to characterize the onto linear isometries of the weighted Bergman spaces over balls and polydiscs, (all isometries referred to are assumed to be linear).2. Preliminaries. Horowitz [3] first defined the weighted Bergman space Ap,α(0 < p < ∞, 0 < α < ∞) to be the space of holomorphic functions f in the disc which satisfy(1)

scholarly journals The Berezin Transform of Toeplitz Operators on the Weighted Bergman Space

2021 ◽  
Author(s):  
Cezhong Tong ◽  
Junfeng Li ◽  
Hicham Arroussi
Keyword(s):  
Bergman Space ◽  
Toeplitz Operators ◽  
Reproducing Kernel ◽  
Bergman Spaces ◽  
Essential Norm ◽  
Berezin Transform ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Kernel Estimates ◽  
Berezin Transforms

AbstractIn this paper, we obtain some interesting reproducing kernel estimates and some Carleson properties that play an important role. We characterize the bounded and compact Toeplitz operators on the weighted Bergman spaces with Békollé-Bonami weights in terms of Berezin transforms. Moreover, we estimate the essential norm of them assuming that they are bounded.

scholarly journals (m,λ)-Berezin Transform on the Weighted Bergman Spaces over the Polydisk

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Ran Li ◽  
Yufeng Lu
Keyword(s):  
Linear Operator ◽  
Bergman Space ◽  
Bounded Linear Operator ◽  
Toeplitz Operators ◽  
Bergman Spaces ◽  
Main Tool ◽  
Berezin Transform ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Bounded Linear

We prove that every bounded linear operator on weighted Bergman space over the polydisk can be approximated by Toeplitz operators under some conditions. The main tool here is the so-called(m,λ)-Berezin transform. In particular, our results generalized the results of K. Nam and D. C. Zheng to the case of operators acting onAλ2(Dn).

scholarly journals Volterra composition operators from generalized weighted weighted Bergman spaces toµ-Bloch spaces

2009 ◽  
Vol 7 (3) ◽  
pp. 225-240 ◽  
Author(s):  
Xiangling Zhu
Keyword(s):  
Holomorphic Function ◽  
Unit Ball ◽  
Bergman Space ◽  
Composition Operators ◽  
Bloch Space ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Bloch Spaces

Letφbe a holomorphic self-map andgbe a fixed holomorphic function on the unit ballB. The boundedness and compactness of the operatorTg,φf(z)=∫01f(φ(tz))ℜg(tz)dttfrom the generalized weighted Bergman space into the µ-Bloch space are studied in this paper.

GENERALISED WEIGHTED COMPOSITION OPERATORS ON BERGMAN SPACES INDUCED BY DOUBLING WEIGHTS

2021 ◽  
pp. 1-13
Author(s):  
BIN LIU
Keyword(s):  
Bergman Space ◽  
Lebesgue Space ◽  
Borel Measure ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Weighted Composition Operators ◽  
Doubling Weights ◽  
Weighted Composition

Abstract We characterise bounded and compact generalised weighted composition operators acting from the weighted Bergman space $A^p_\omega $ , where $0<p<\infty $ and $\omega $ belongs to the class $\mathcal {D}$ of radial weights satisfying a two-sided doubling condition, to a Lebesgue space $L^q_\nu $ . On the way, we establish a new embedding theorem on weighted Bergman spaces $A^p_\omega $ which generalises the well-known characterisation of the boundedness of the differentiation operator $D^n(f)=f^{(n)}$ from the classical weighted Bergman space $A^p_\alpha $ to the Lebesgue space $L^q_\mu $ , induced by a positive Borel measure $\mu $ , to the setting of doubling weights.

scholarly journals On the exact value of the norm of the Hilbert matrix operator on weighted Bergman spaces

2021 ◽  
Vol 46 (1) ◽  
pp. 201-224
Author(s):  
Mikael Lindström ◽  
Santeri Miihkinen ◽  
Niklas Wikman
Keyword(s):  
Bergman Spaces ◽  
Matrix Operator ◽  
Weighted Bergman Spaces ◽  
Hilbert Matrix

scholarly journals Linear functionals on some weighted Bergman spaces

1990 ◽  
Vol 42 (3) ◽  
pp. 417-425 ◽  
Author(s):  
Maher M.H. Marzuq
Keyword(s):  
Bergman Space ◽  
Analytic Functions ◽  
Unit Disc ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Linear Functionals

The weighted Bergman space Ap, α, 0 < p < 1, a > −1 of analytic functions on the unit disc Δ in C is an F-space. We determine the dual of Ap, α explicitly.

scholarly journals Extension of Zhu's solution to Lotto's conjecture on the weighted Bergman spaces

2004 ◽  
Vol 2004 (41) ◽  
pp. 2199-2203
Author(s):  
Abebaw Tadesse
Keyword(s):  
Hardy Space ◽  
Bergman Space ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces

We reformulate Lotto's conjecture on the weighted Bergman spaceAα2setting and extend Zhu's solution (on the Hardy spaceH2) to the spaceAα2.

scholarly journals Addendum to the Paper “A note on Weighted bergman Spaces and the cesaro operator”

2005 ◽  
Vol 180 ◽  
pp. 77-90 ◽  
Author(s):  
Der-Chen Chang ◽  
Stevo Stević
Keyword(s):  
Bergman Space ◽  
Bergman Spaces ◽  
Holomorphic Functions ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Cesàro Operator ◽  
Integral Means ◽  
Measurable Functions ◽  
Unit Polydisk ◽  
Cesaro Operator

AbstractLet H(Dn) be the space of holomorphic functions on the unit polydisk Dn, and let , where p, q> 0, α = (α1,…,αn) with αj > -1, j =1,..., n, be the class of all measurable functions f defined on Dn such thatwhere Mp(f,r) denote the p-integral means of the function f. Denote the weighted Bergman space on . We provide a characterization for a function f being in . Using the characterization we prove the following result: Let p> 1, then the Cesàro operator is bounded on the space .

scholarly journals Compact Differences of Composition Operators on Bergman Spaces Induced by Doubling Weights

2021 ◽  
Author(s):  
Bin Liu ◽  
Jouni Rättyä ◽  
Fanglei Wu
Keyword(s):  
Bergman Space ◽  
Open Problem ◽  
Lebesgue Space ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Carleson Measures ◽  
Doubling Weights ◽  
The Way

AbstractBounded and compact differences of two composition operators acting from the weighted Bergman space $$A^p_\omega $$ A ω p to the Lebesgue space $$L^q_\nu $$ L ν q , where $$0<q<p<\infty $$ 0 < q < p < ∞ and $$\omega $$ ω belongs to the class "Equation missing" of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proofs a new description of q-Carleson measures for $$A^p_\omega $$ A ω p , with $$p>q$$ p > q and "Equation missing", involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q-Carleson measures for the classical weighted Bergman space $$A^p_\alpha $$ A α p with $$-1<\alpha <\infty $$ - 1 < α < ∞ to the setting of doubling weights. The case "Equation missing" is also briefly discussed and an open problem concerning this case is posed.

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