Dual Toeplitz operators on orthogonal complement of the harmonic Dirichlet space

2016 ◽  
Vol 33 (3) ◽  
pp. 383-402 ◽  
Author(s):  
Yong Chen ◽  
Tao Yu ◽  
Yi Le Zhao
Keyword(s):  
Toeplitz Operators ◽  
Dirichlet Space ◽  
Orthogonal Complement ◽  
Harmonic Dirichlet Space

scholarly journals Properties of Commutativity of Dual Toeplitz Operators on the Orthogonal Complement of Pluriharmonic Dirichlet Space over the Ball

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Yinyin Hu ◽  
Yufeng Lu ◽  
Tao Yu
Keyword(s):  
Sobolev Space ◽  
Unit Ball ◽  
Toeplitz Operators ◽  
Dirichlet Space ◽  
Orthogonal Complement ◽  
Pluriharmonic Functions

We completely characterize the pluriharmonic symbols for (semi)commuting dual Toeplitz operators on the orthogonal complement of the pluriharmonic Dirichlet space in Sobolev space of the unit ball. We show that, forfandgpluriharmonic functions,SfSg=SgSfon(Dh)⊥if and only iffandgsatisfy one of the following conditions:(1)bothfandgare holomorphic;(2)bothf¯andg¯are holomorphic;(3)there are constantsαandβ, both not being zero, such thatαf+βgis constant.

scholarly journals Hyponormal Toeplitz Operators on the Dirichlet Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Puyu Cui ◽  
Yufeng Lu
Keyword(s):  
Toeplitz Operators ◽  
Dirichlet Space ◽  
Dirichlet Spaces ◽  
Harmonic Dirichlet Space

We completely characterize the hyponormality of bounded Toeplitz operators with Sobolev symbols on the Dirichlet space and the harmonic Dirichlet space.

Commuting dual Toeplitz operators on the harmonic Dirichlet space

2016 ◽  
Vol 32 (9) ◽  
pp. 1099-1105 ◽  
Author(s):  
Jing Yu Yang ◽  
Yin Yin Hu ◽  
Yu Feng Lu ◽  
Tao Yu
Keyword(s):  
Toeplitz Operators ◽  
Dirichlet Space ◽  
Harmonic Dirichlet Space

Bergman Spaces on Disconnected Domains

1996 ◽  
Vol 48 (2) ◽  
pp. 225-243
Author(s):  
Alexandru Aleman ◽  
Stefan Richter ◽  
William T. Ross
Keyword(s):  
Measure Zero ◽  
Bergman Spaces ◽  
Dirichlet Space ◽  
Invariant Subspaces ◽  
Compact Set ◽  
Bounded Region ◽  
Area Measure ◽  
Weak Star ◽  
Harmonic Dirichlet Space ◽  
Disconnected Domains

AbstractFor a bounded region G ⊂ ℂ and a compact set K ⊂ G, with area measure zero, we will characterize the invariant subspaces ℳ (under ƒ → zƒ) of the Bergman space (G \ K), 1 ≤ p < ∞, which contain (G) and with dim(ℳ/(z - λ)ℳ) = 1 for all λ ∈ G \ K. When G \ K is connected, we will see that dim(ℳ/(z - λ)ℳ) = 1 for all λ ∈ G \ K and thus in this case we will have a complete description of the invariant subspaces lying between (G) and (G \ K). When p = ∞, we will remark on the structure of the weak-star closed z-invariant subspaces between H∞(G) and H∞(G \ K). When G \ K is not connected, we will show that in general the invariant subspaces between (G) and (G \ K) are fantastically complicated. As an application of these results, we will remark on the complexity of the invariant subspaces (under ƒ → ζƒ) of certain Besov spaces on K. In particular, we shall see that in the harmonic Dirichlet space , there are invariant subspaces ℱ such that the dimension of ζℱ in ℱ is infinite.

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