Weighted composition operators on weighted Bergman spaces induced by doubling weights

2020 ◽  
Vol 126 (3) ◽  
pp. 519-539
Author(s):  
Juntao Du ◽  
Songxiao Li ◽  
Yecheng Shi
Keyword(s):  
Composition Operators ◽  
Bergman Spaces ◽  
Essential Norm ◽  
Weighted Bergman Spaces ◽  
Weighted Composition Operators ◽  
Schatten Class ◽  
Doubling Weight ◽  
Doubling Weights ◽  
Type Spaces ◽  
Weighted Composition

In this paper, we investigate the boundedness, compactness, essential norm and the Schatten class of weighted composition operators $uC_\varphi $ on Bergman type spaces $A_\omega ^p $ induced by a doubling weight ω. Let $X=\{u\in H(\mathbb{D} ): uC_\varphi \colon A_\omega ^p\to A_\omega ^p\ \text {is bounded}\}$. For some regular weights ω, we obtain that $X=H^\infty $ if and only if ϕ is a finite Blaschke product.

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 Weighted Composition Operators from Weighted Bergman Spaces to Weighted-Type Spaces on the Upper Half-Plane

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Stevo Stević ◽  
Ajay K. Sharma ◽  
S. D. Sharma
Keyword(s):  
Half Plane ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Weighted Composition Operators ◽  
Type Spaces ◽  
Weighted Composition ◽  
Upper Half Plane ◽  
Weighted Type

Letψbe a holomorphic mapping on the upper half-planeΠ+={z∈ℂ:Jz>0}andφbe a holomorphic self-map ofΠ+. We characterize bounded weighted composition operators acting from the weighted Bergman space to the weighted-type space on the upper half-plane. Under a mild condition onψ, we also characterize the compactness of these operators.

scholarly journals Weighted composition operators with closed range

2007 ◽  
Vol 75 (3) ◽  
pp. 331-354 ◽  
Author(s):  
N. Palmberg
Keyword(s):  
Infinite Order ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Sufficient Conditions ◽  
Weighted Bergman Spaces ◽  
Weighted Composition Operators ◽  
Closed Range ◽  
Type Spaces ◽  
Weighted Composition ◽  
Necessary And Sufficient

We study the closed range property of weighted composition operators on weighted Bergman spaces of infinite order (including the Hardy space of infinite order). We give some necessary and sufficient conditions and find a complete characterisation for weighted composition operators associated with conformal mappings. We also give the corresponding results for composition operators on the Bloch-type spaces. Therefore, the results obtained in this paper also improve and generalise the results of Ghatage, Yan, Zheng and Zorboska.

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