scholarly journals Toeplitz Operators on Dirichlet-Type Space of Unit Ball

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Jin Xia ◽  
Xiaofeng Wang ◽  
Guangfu Cao
Keyword(s):  
Toeplitz Operator ◽  
Boundary Point ◽  
Toeplitz Operators ◽  
Radial Function ◽  
Type Space ◽  
Dirichlet Type ◽  
Algebraic Properties ◽  
Commuting Problem ◽  
Product Problem ◽  
Dirichlet Type Space

We construct a functionuinL2Bn, dVwhich is unbounded on any neighborhood of each boundary point ofBnsuch that Toeplitz operatorTuis a Schattenp-class0<p<∞operator on Dirichlet-type spaceDBn, dV. Then, we discuss some algebraic properties of Toeplitz operators with radial symbols on the Dirichlet-type spaceDBn, dV. We determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator. We investigate the zero-product problem for several Toeplitz operators with radial symbols. Furthermore, the corresponding commuting problem of Toeplitz operators whose symbols are of the formξkuis studied, wherek ∈ Zn,ξ ∈ ∂Bn, anduis a radial function.

scholarly journals Algebraic Properties of Toeplitz Operators on the Pluriharmonic Bergman Space

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Jingyu Yang ◽  
Liu Liu ◽  
Yufeng Lu
Keyword(s):  
Toeplitz Operator ◽  
Bergman Space ◽  
Finite Rank ◽  
Toeplitz Operators ◽  
Sufficient Conditions ◽  
Rank Product ◽  
Algebraic Properties ◽  
Necessary And Sufficient ◽  
Product Problem ◽  
Quasihomogeneous Symbols

We study some algebraic properties of Toeplitz operators with radial or quasihomogeneous symbols on the pluriharmonic Bergman space. We first give the necessary and sufficient conditions for the product of two Toeplitz operators with radial symbols to be a Toeplitz operator and discuss the zero-product problem for several Toeplitz operators with radial symbols. Next, we study the finite-rank product problem of several Toeplitz operators with quasihomogeneous symbols. Finally, we also investigate finite rank commutators and semicommutators of two Toeplitz operators with quasihomogeneous symbols.

scholarly journals Algebraic Properties of Quasihomogeneous and Separately Quasihomogeneous Toeplitz Operators on the Pluriharmonic Bergman Space

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Hongyan Guan ◽  
Liu Liu ◽  
Yufeng Lu
Keyword(s):  
Unit Ball ◽  
Toeplitz Operator ◽  
Bergman Space ◽  
Trivial Solution ◽  
Toeplitz Operators ◽  
The Other ◽  
Homogeneous Functions ◽  
Algebraic Properties ◽  
Product Problem

We study some algebraic properties of Toeplitz operator with quasihomogeneous or separately quasihomogeneous symbol on the pluriharmonic Bergman space of the unit ball inℂn. We determine when the product of two Toeplitz operators with certain separately quasi-homogeneous symbols is a Toeplitz operator. Next, we discuss the zero-product problem for several Toeplitz operators, one of whose symbols is separately quasihomogeneous and the others are quasi-homogeneous functions, and show that the zero-product problem for two Toeplitz operators has only a trivial solution if one of the symbols is separately quasihomogeneous and the other is arbitrary. Finally, we also characterize the commutativity of certain quasihomogeneous or separately quasihomogeneous Toeplitz operators.

scholarly journals Algebraic Properties of Toeplitz Operators on the Pluri-harmonic Fock Space

2017 ◽  
Vol 9 (6) ◽  
pp. 67 ◽  
Author(s):  
Dieudonne Agbor
Keyword(s):  
Toeplitz Operator ◽  
Finite Rank ◽  
Toeplitz Operators ◽  
Fock Space ◽  
Algebraic Properties ◽  
Product Problem

We study some algebraic properties of Toeplitz operators  with radial and quasi homogeneous symbols on the pluriharmonic Fock space over $\mathbb{C}^{n}$. We determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator, the zero-product problem for the product of two Toeplitz operators.  Next we characterize the commutativity of Toeplitz operators with quasi homogeneous symbols and finally we study finite rank of the product of Toeplitz operators with quasi homogeneous symbols.

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 Characterizations of a Class of Dirichlet-Type Spaces and Related Operators

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Xiaosong Liu ◽  
Songxiao Li
Keyword(s):  
Boundary Value ◽  
Type Space ◽  
Dirichlet Type ◽  
Type Spaces ◽  
Dirichlet Type Spaces ◽  
Dirichlet Type Space

In this paper, some characterizations are given in terms of the boundary value and Poisson extension for the Dirichlet-type space D μ . The multipliers of D μ and Hankel-type operators from D μ to L 2 P μ d A are also investigated.

scholarly journals Algebraic Properties of Toeplitz Operators on the Polydisk

2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Bo Zhang ◽  
Yanyue Shi ◽  
Yufeng Lu
Keyword(s):  
Bergman Space ◽  
Finite Rank ◽  
Toeplitz Operators ◽  
Rank Product ◽  
Algebraic Properties ◽  
Product Problem ◽  
Quasihomogeneous Symbols

We discuss some algebraic properties of Toeplitz operators on the Bergman space of the polydiskDn. Firstly, we introduce Toeplitz operators with quasihomogeneous symbols and property (P). Secondly, we study commutativity of certain quasihomogeneous Toeplitz operators and commutators of diagonal Toeplitz operators. Thirdly, we discuss finite rank semicommutators and commutators of Toeplitz operators with quasihomogeneous symbols. Finally, we solve the finite rank product problem for Toeplitz operators on the polydisk.

Characterizations of the Dirichlet-Type Space

2014 ◽  
Vol 9 (6) ◽  
pp. 1269-1286 ◽  
Author(s):  
Xiaosong Liu ◽  
Gerardo R. Chacón ◽  
Zengjian Lou
Keyword(s):  
Type Space ◽  
Dirichlet Type ◽  
Dirichlet Type Space
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