Equivalent Characterizations and Pointwise Multipliers of Normal Weight Dirichlet Space on the Unit Ball

2021 ◽  
Vol 15 (2) ◽  
Author(s):  
Meijuan Tan ◽  
Xuejun Zhang
Keyword(s):  
Unit Ball ◽  
Dirichlet Space ◽  
Normal Weight ◽  
Pointwise Multipliers

scholarly journals ESSENTIAL NORM OF EXTENDED CESÀRO OPERATORS FROM ONE BERGMAN SPACE TO ANOTHER

2011 ◽  
Vol 85 (2) ◽  
pp. 307-314 ◽  
Author(s):  
ZHANGJIAN HU
Keyword(s):  
Unit Ball ◽  
Bergman Space ◽  
Holomorphic Functions ◽  
Essential Norm ◽  
Normal Weight ◽  
Cesàro Operator ◽  
Radial Derivative ◽  
Image Position ◽  
Cesaro Operator

AbstractLet Ap(φ) be the pth Bergman space consisting of all holomorphic functions f on the unit ball B of ℂn for which $\|f\|^p_{p,\varphi }= \int _B |f(z)|^p \varphi (z) \,dA(z)\lt +\infty $, where φ is a given normal weight. Let Tg be the extended Cesàro operator with holomorphic symbol g. The essential norm of Tg as an operator from Ap (φ) to Aq (φ) is denoted by $\|T_g\|_{e, A^p (\varphi )\to A^q (\varphi )} $. In this paper it is proved that, for p≤q, with 1/k=(1/p)−(1/q) , where ℜg(z) is the radial derivative of g; and for p>q, with 1/s=(1/q)−(1/p) .

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.

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 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 On an Integral-Type Operator Acting between Bloch-Type Spaces on the Unit Ball

2010 ◽  
Vol 2010 ◽  
pp. 1-14 ◽  
Author(s):  
Stevo Stević ◽  
Sei-Ichiro Ueki
Keyword(s):  
Holomorphic Function ◽  
Weight Function ◽  
Unit Ball ◽  
Normal Weight ◽  
Type Operator ◽  
Open Unit ◽  
Integral Type ◽  
Open Unit Ball ◽  
Radial Derivative ◽  
Type Spaces

Let𝔹denote the open unit ball ofℂn. For a holomorphic self-mapφof𝔹and a holomorphic functiongin𝔹withg(0)=0, we define the following integral-type operator:Iφgf(z)=∫01ℜf(φ(tz))g(tz)(dt/t),z∈𝔹. Hereℜfdenotes the radial derivative of a holomorphic functionfin𝔹. We study the boundedness and compactness of the operator between Bloch-type spacesℬωandℬμ, whereωis a normal weight function andμis a weight function. Also we consider the operator between the little Bloch-type spacesℬω,0andℬμ,0.

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.

Sign in / Sign up

Export Citation Format

Share Document