scholarly journals Integration Operator Acting on Hardy and Weighted Bergman Spaces

2007 ◽  
Vol 75 (3) ◽  
pp. 431-446 ◽  
Author(s):  
Jouni Rättyä
Keyword(s):  
Analytic Function ◽  
Hardy Space ◽  
Asymptotic Formula ◽  
Unit Disc ◽  
Bergman Spaces ◽  
Essential Norm ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Integration Operator ◽  
Complete Characterisation

Questions related to the operator Jg(f)(z):= ∫xof (ζ)g′(ζ) dζ, induced by an analytic function g in the unit disc, are studied. It is shown that a function G is the derivative of a function in the Hardy space Hp if and only if it is of the form G = Fψ′ where F ∈ Hq, ψ ∈ H3 and 1/s = 1/p − 1/q. Moreover, a complete characterisation of when Jg is bounded or compact from one weighted Bergman space into another is established, and an asymptotic formula for the essential norm of Jg, the distance from compact operators in the operator norm, is given. As an immediate consequence it is obtained that if p < 2 + α and α > −1, then any primitive of belongs to where q = ((2 + α) p)/(2 + α − p). For α = −1 this is a sharp result by Hardy and Littlewood on primitives of functions in Hardy space , 0 < p < 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 Compact Toeplitz operators with continuous symbols on weighted Bergman spaces

2000 ◽  
Vol 42 (1) ◽  
pp. 31-35 ◽  
Author(s):  
Takahiko Nakazi ◽  
Rikio Yoneda
Keyword(s):  
Continuous Function ◽  
Unit Disc ◽  
Toeplitz Operators ◽  
Borel Measure ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Open Unit ◽  
Open Unit Disc ◽  
Finite Borel Measure

Let L^2_a (D, d\sigma d\theta /2\pi ) be a complete weighted Bergman space on the open unit disc D, where d\sigma is a positive finite Borel measure on [0, 1). We show the following : when \phi is a continuous function on the closed unit disc \bar {D}, T_\phi is compact if and only if \phi = 0 on \partial D.1991 Mathematics Subject Classification 47B35, 47B07.

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 Composition operators on weighted Bergman spaces of a half-plane

2011 ◽  
Vol 54 (2) ◽  
pp. 373-379 ◽  
Author(s):  
Sam J. Elliott ◽  
Andrew Wynn
Keyword(s):  
Spectral Radius ◽  
Half Plane ◽  
Composition Operator ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Essential Norm ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Angular Derivative ◽  
The Right

AbstractWe use induction and interpolation techniques to prove that a composition operator induced by a map ϕ is bounded on the weighted Bergman space $\mathcal{A}^2_\alpha(\mathbb{H})$ of the right half-plane if and only if ϕ fixes the point at ∞ non-tangentially and if it has a finite angular derivative λ there. We further prove that in this case the norm, the essential norm and the spectral radius of the operator are all equal and are given by λ(2+α)/2.

Hankel Operators on Pseudoconvex Domains of Finite Type in ℂ2

1998 ◽  
Vol 50 (3) ◽  
pp. 658-672 ◽  
Author(s):  
Frédéric Symesak
Keyword(s):  
Hardy Space ◽  
Finite Type ◽  
Pseudoconvex Domain ◽  
Bergman Spaces ◽  
Hankel Operators ◽  
Weighted Bergman Spaces ◽  
Pseudoconvex Domains ◽  
Sufficient Condition ◽  
Schatten Class ◽  
Strictly Pseudoconvex Domain

AbstractThe aimof this paper is to study small Hankel operators h on the Hardy space or on weighted Bergman spaces,where Ω is a finite type domain in ℂ2 or a strictly pseudoconvex domain in ℂn. We give a sufficient condition on the symbol ƒ so that h belongs to the Schatten class Sp, 1 ≤ p < +∞.

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