Dual Toeplitz Operators On The Orthogonal Complement Of The Dirichlet Space

The operator theory on function spaces is one of the significant branch in functional analysis.In this dissertation,we mainly study the dual Toeplitz operaors on the orthog-onal complement of the Dirichlet space and Larger Dirichlet space.The commutativity, productness and algebraic properties of the dual Toeplitz operaors S_φare mainly con-cerned.It completed by the advantage of the tight ralations between Toeplitz operaors T_φ,Hankel operators H_φ,and the operators R_φ.In Chapter 1,we summarized related research ground of the Toeplitz operaors T_φand dual Toeplitz operaors S_φand show the significance of the research.In Chapter 2,we characterize commuting dual Toeplitz operaors with harmonic symbols and the dual Toeplitz operaors which commute with a nonconstant analytic dual Toeplitz operaors on the orthogonal complement of the Dirichlet space D₀,we also investigate the product of dual Toeplitz operaors and the finite rank perturbation of the product of dual Toeplitz operaors,obtain the following results:Theorem 2.3.1 Suppose thatφ,ψ∈W^1,∞(D)be harmonic functions,then S_φS_ψ= S_ψS_φif and only if one of the following statements holds:(â…°)Bothφandψare holomorphic on D;(â…±)Bothφandψare antiholomorphic on D;(â…²)A nontrivial linear combination ofφandψis constant on D.Theorem 2.4.1 Suppose thatφis a nonconstant bounded analytic function andψ∈W^1,∞(D)such that S_φand S_ψcommute,thenψis analytic. Theorem 2.5.1 Suppose thatφ,ψ∈W^1,∞(D)be harmonic functions,then S_φS_ψ= S_φψif and only if one of the following statements holds:(â…°)φis holomorphic on D;(â…±)ψis antiholomorphic on D.Theorem 2.5.6 Suppose thatφ,ψ∈W^1,∞(D),and S_φS_ψis a finite rank pertur-bation of dual Toeplitz operator S_u,for some u∈W^1,∞(D),thenφψ= u.In Chapter 3,we also characterize commuting dual Toeplitz operaors with harmonic symbols and the dual Toeplitz operaors which commute with a nonconstant analytic dual Toeplitz operaors on the orthogonal complement of the Larger Dirichlet space D, in addition,we investigate the product of dual Toeplitz operaors and the finite rank perturbation of the product of dual Toeplitz operaors,and obtain the following results:Theorem 3.3.1 Suppose thatφ,ψ∈W^1,∞(D)be harmonic functions,then S_φS_ψ= S_ψS_φif and only if one of the following statements holds:(â…°)Bothφandψare holomorphic on D;(â…±)A nontrivial linear combination ofφandψis constant on D.Theorem 3.4.1 Suppose that F is a nonconstant bounded analytic function andψ∈W^1,∞(D)such that S_φand S_ψcommute,thenψis analytic.Theorem 3.5.1 Suppose thatφ,ψ∈W^1,∞(D)be harmonic functions,then S_φS_ψ= S_φψif and only if one of the following statements holds:(â…°)φis holomorphic on D;(â…±)ψis a constant on D.Theorem 3.5.6 Suppose thatφ,ψ∈W^1,∞(D),and S_φS_ψis a finite rank pertur-bation of dual Toeplitz operator S_u for some u∈L^∞,1(D),thenφψ= u.In Chapter 4,we investigate the compactness of the operaor R_φon the orthogonal complement of the Dirichlet space D₀,and obtain the following results: Theorem 4.3.1 For anyφ∈C¹((?)),R_φis a compact operator from D₀^⊥to D₀.Theorem 4.3.2 For anyφ,ψ∈C¹((?)),S_φψ-S_φS_ψ∈K(D₀⊥).(K(D₀^⊥)denotes the compact operator algebra on D₀^⊥.)...