scholarly journals Reducing subspaces for analytic multipliers of the Bergman space

2012 ◽  
Vol 263 (6) ◽  
pp. 1744-1765 ◽  
Author(s):  
Ronald G. Douglas ◽  
Mihai Putinar ◽  
Kai Wang
Keyword(s):  
Bergman Space ◽  
Reducing Subspaces

scholarly journals Reducing Subspaces of Some Multiplication Operators on the Bergman Space over Polydisk

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Yanyue Shi ◽  
Na Zhou
Keyword(s):  
Bergman Space ◽  
Multiplication Operators ◽  
Direct Sum ◽  
Reducing Subspace ◽  
Reducing Subspaces

We consider the reducing subspaces ofMzNonAα2(Dk), wherek≥3,zN=z1N1⋯zkNk, andNi≠Njfori≠j. We prove that each reducing subspace ofMzNis a direct sum of some minimal reducing subspaces. We also characterize the minimal reducing subspaces in the cases thatα=0andα∈(-1,+∞)∖Q, respectively. Finally, we give a complete description of minimal reducing subspaces ofMzNonAα2(D3)withα>-1.

scholarly journals Reducing subspaces for multiplication operators on the Dirichlet space through local inverses and Riemann surfaces

2017 ◽  
Vol 4 (1) ◽  
pp. 84-119 ◽  
Author(s):  
Caixing Gu ◽  
Shuaibing Luo ◽  
Jie Xiao
Keyword(s):  
Bergman Space ◽  
Blaschke Product ◽  
Riemann Surfaces ◽  
Dirichlet Space ◽  
Multiplication Operator ◽  
Multiplication Operators ◽  
Finite Blaschke Product ◽  
Full Characterization ◽  
Reducing Subspaces

AbstractThis paper gives a full characterization of the reducing subspaces for the multiplication operator Mϕ on the Dirichlet space with symbol of finite Blaschke product ϕ of order 5I 6I 7. The reducing subspaces of Mϕ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surfaces to study the reducing subspaces of Mϕ on the Bergman space. By this means, we determine the reducing subspaces of Mϕ on the Dirichlet space and answer some questions of Douglas-Putinar-Wang in [6].

scholarly journals On Similarity and Reducing Subspaces of the n-Shift plus Certain Weighted Volterra Operator

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Yucheng Li ◽  
Hao Chen ◽  
Wenhua Lan
Keyword(s):  
Hilbert Space ◽  
Bergman Space ◽  
Volterra Operator ◽  
Multiplication Operator ◽  
Degree Polynomial ◽  
Reducing Subspaces

Let g(z) be an n-degree polynomial (n≥2). Inspired by Sarason’s result, we introduce the operator T1 defined by the multiplication operator Mg plus the weighted Volterra operator Vg on the Bergman space. We show that the operator T1 is similar to Mg on some Hilbert space Sg2(D). Then for g(z)=zn, by using matrix manipulations, the reducing subspaces of the corresponding operator T2 on the Bergman space are characterized.

scholarly journals Classification of Reducing Subspaces of a Class of Multiplication Operators on the Bergman Space via the Hardy Space of the Bidisk

2010 ◽  
Vol 62 (2) ◽  
pp. 415-438 ◽  
Author(s):  
Shunhua Sun ◽  
Dechao Zheng ◽  
Changyong Zhong
Keyword(s):  
Hardy Space ◽  
Unit Disk ◽  
Bergman Space ◽  
Blaschke Product ◽  
Multiplication Operator ◽  
Multiplication Operators ◽  
Reducing Subspaces

AbstractIn this paper we obtain a complete description of nontrivial minimal reducing subspaces of the multiplication operator by a Blaschke product with four zeros on the Bergman space of the unit disk via the Hardy space of the bidisk.

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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