Riccati equations and Toeplitz-Berezin type symbols on Dirichlet space of unit ball

2017 ◽  
Vol 12 (4) ◽  
pp. 769-785
Author(s):  
Jianjun Chen ◽  
Xiaofeng Wang ◽  
Jin Xia ◽  
Guangfu Cao
Keyword(s):  
Unit Ball ◽  
Dirichlet Space ◽  
Riccati Equations

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 Extreme points in spaces between Dirichlet and Vanishing Mean Oscillation

2003 ◽  
Vol 67 (3) ◽  
pp. 365-375
Author(s):  
K. J. Wirths ◽  
J. Xiao
Keyword(s):  
Unit Ball ◽  
Unit Circle ◽  
Dirichlet Space ◽  
Extreme Points ◽  
Closed Unit Ball ◽  
Mean Oscillation ◽  
Image Position ◽  
Complex Valued

For p ∈ (0, ∞) define Qp0(∂Δ) as the space of all Lebesgue measurable complex-valued functions f; on the unit circle ∂Δ for which ∫∂Δf;(z)|dz|/(2π) = 0 andas the open subarc I of ∂Δ varies. Note that each Qp,0(∂Δ) lies between the Dirichlet space and Sarason's vanishing mean oscillation space. This paper determines the extreme points of the closed unit ball of Qp,0(∂Δ) equipped with an appropriate norm.

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.

scholarly journals Norm and Essential Norm of an Integral-Type Operator from the Dirichlet Space to the Bloch-Type Space on the Unit Ball

2010 ◽  
Vol 2010 ◽  
pp. 1-9 ◽  
Author(s):  
Stevo Stević
Keyword(s):  
Unit Ball ◽  
Dirichlet Space ◽  
Essential Norm ◽  
Operator Norm ◽  
Type Operator ◽  
Integral Type ◽  
Type Space ◽  
Bloch Type Space

Operator norm and essential norm of an integral-type operator, recently introduced by this author, from the Dirichlet space to the Bloch-type space on the unit ball in are calculated here.

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