Algebraic Properties of Toeplitz Operators on the Harmonic Dirichlet Space

2010 ◽  
Vol 69 (2) ◽  
pp. 183-201 ◽  
Author(s):  
Yong Chen ◽  
Young Joo Lee ◽  
Quang Dieu Nguyen
Keyword(s):  
Toeplitz Operators ◽  
Dirichlet Space ◽  
Algebraic Properties ◽  
Harmonic Dirichlet Space

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

scholarly journals Toeplitz Operators on the Dirichlet Space of𝔹n

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
HongZhao Lin ◽  
YuFeng Lu
Keyword(s):  
Unit Ball ◽  
Toeplitz Operator ◽  
Toeplitz Operators ◽  
Dirichlet Space ◽  
Algebraic Properties ◽  
Commuting Problem

We study the algebraic properties of Toeplitz operators on the Dirichlet space of the unit ball𝔹n. We characterize pluriharmonic symbol for which the corresponding Toeplitz operator is normal or isometric. We also obtain descriptions of conjugate holomorphic symbols of commuting Toeplitz operators. Finally, the commuting problem of Toeplitz operators whose symbols are of the formzpz¯qϕ(|z|2)is studied.

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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