Corona Theorem for the Dirichlet-Type Space

2022 ◽  
Vol 32 (3) ◽  
Author(s):  
Shuaibing Luo
Keyword(s):  
Corona Theorem ◽  
Type Space ◽  
Dirichlet Type ◽  
Dirichlet Type Space

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 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.

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

scholarly journals The essential norm of a composition operator mapping intoQktype spaces

2008 ◽  
Vol 6 (3) ◽  
pp. 241-258 ◽  
Author(s):  
Marko Kotilainen ◽  
Jouni Rättyä
Keyword(s):  
Asymptotic Formula ◽  
Composition Operator ◽  
Unit Disc ◽  
Bloch Space ◽  
Integral Condition ◽  
Essential Norm ◽  
Type Space ◽  
Operator Mapping ◽  
Dirichlet Type ◽  
Dirichlet Type Space

An asymptotic formula for the essential norm of the composition operatorCφ(f):=f∘φ, induced by an analytic self-mapφof the unit disc, mapping from theα-Bloch spaceℬαor the Dirichlet type spaceDαpintoQk(p,q)is established in terms of an integral condition.

QpSpaces and Dirichlet Type Spaces

2017 ◽  
Vol 60 (4) ◽  
pp. 690-704 ◽  
Author(s):  
Guanlong Bao ◽  
Nihat Gökhan Gögüs ◽  
Stamatis Pouliasis
Keyword(s):  
Hilbert Spaces ◽  
Invariant Function ◽  
Open Unit ◽  
Sufficient Condition ◽  
Borel Measures ◽  
Dirichlet Type ◽  
Type Spaces ◽  
Dirichlet Type Spaces ◽  
Decomposition Theorems ◽  
Dirichlet Type Space

AbstractIn this paper, we show that the Möbius invariant function space Qpcan be generated by variant Dirichlet type spaces 𝒟μ,pinduced by finite positive Borel measures μ on the open unit disk. A criterion for the equality between the space 𝒟μ,pand the usual Dirichlet type space 𝒟pis given. We obtain a sufficient condition to construct different 𝒟μ,pspaces and provide examples. We establish decomposition theorems for 𝒟μ,pspaces and prove that the non-Hilbert space Qpis equal to the intersection of Hilbert spaces 𝒟μ,p. As an application of the relation between Qpand 𝒟μ,pspaces, we also obtain that there exist different 𝒟μ,pspaces; this is a trick to prove the existence without constructing examples.

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