Commuting Dual Toeplitz Operators on the Polydisk

We completely characterize commutativity of S and Sψ on La2(Dn)⊥ for bounded pluriharmonic symbols and ψ on Dn, and prove that SSψ = Sψ if and only if is analytic or ψˉ is analytic.

Block dual Toeplitz Operators on the Orthogonal Complements of Fock Spaces

In this paper,we characterize the boundedness and compactness of the block dual Toeplitz operators acting on the orthogonal complement of the Fock spaces,and the compactness of the finite sum of two dual Toeplitz operators products.The commutator and semi-commutator induced by the block dual Toeplitz operators are considered.

Commuting Dual Toeplitz Operators on the Polydisk

On the polydisk, the commutativity of dual Toeplitz operators is studied. We obtain characterizations of commuting dual Toeplitz operators, essentially commuting dual Toeplitz operators and essentially semi-commuting dual Toeplitz operators.

SUMS OF DUAL TOEPLITZ PRODUCTS ON THE ORTHOGONAL COMPLEMENTS OF FOCK-SOBOLEV SPACES

We consider dual Toeplitz operators on the orthogonal complements of the FockSobolev spaces of all nonnegative real orders.First,for symbols in a certain class containing all bounded functions,we study the problem of when an operator which is finite sums of the dual Toeplitz products is compact or zero.Next,for bounded symbols,we construct a symbol map and exhibit a short exact sequence associated with the C^(*)-algebra generated by all dual Toeplitz operators with bounded symbols.

Commuting Dual Toeplitz Operators on the Orthogonal Complement of the Dirichlet Space

In this paper we characterize commuting dual Toeplitz operators with harmonic symbols on the orthogonal complement of the Dirichlet space in the Sobolev space. We also obtain the sufficient and necessary conditions for the product of two dual Toeplitz operators with harmonic symbols to be a finite rank perturbation of a dual Toeplitz operator.

Dual Toeplitz Operators on the Sphere

Dual Toeplitz operators on the Hardy space of the unit circle are anti-unitarily equivalent to Toeplitz operators. In higher dimensions, for instance on the unit sphere, dual Toeplitz operators might behave quite differently and, therefore, seem to be a worth studying new class of Toeplitz-type operators. The purpose of this paper is to introduce and start a systematic investigation of dual Toeplitz operators on the orthogonal complement of the Hardy space of the unit sphere in Cn . In particular, we establish a corresponding spectral inclusion theorem and a Brown-Halmos type theorem. On the other hand, we characterize commuting dual Toeplitz operators as well as normal and quasinormal ones.

Algebraic Properties of Dual Toeplitz Operators on the Orthogonal Complement of the Dirichlet Space

In this paper we investigate some algebra properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space in the Sobolev space. We completely characterize commuting dual Toeplitz operators with harmonic symbols, and show that a dual Toeplitz operator commutes with a nonconstant analytic dual Toeplitz operator if and only if its symbol is analytic. We also obtain the sufficient and necessary conditions on the harmonic symbols for SφSφψ= Sφψ.

Commuting Dual Toeplitz Operators on Weighted Bergman Spaces of the Unit Ball

In this paper, we study the commutativity of dual Toeplitz operators on weighted Bergman spaces of the unit ball. We obtain the necessary and sufficient conditions for the commutativity, essential commutativity and essential semi-commutativity of dual Toeplitz operator on the weighted Bergman spaces of the unit ball.

Commuting dual Toeplitz operators on the harmonic Bergman space

We completely characterize commuting dual Toeplitz operators with bounded harmonic symbols on the harmonic Bergman space of the unit disk. We show that for harmonic ■ and ψ, S■Sψ= SψS■ on(L2h)⊥if and only if ■ and ψ satisfy one of the following conditions:(1) Both ■ and ψ are analytic on D.(2) Both ■ and ψ are anti-analytic on D.(3) There exist complex constants α and β, not both 0, such that ■ = αψ + β.Furthermore, we give the necessary and sufficient conditions for S■Sψ= S■ψ.

Dual Toeplitz Operators on Orthogonal Complement of the Harmonic Dirichlet Space

In this paper, we study some algebraic and spectral properties of dual Toeplitz operators on the orthogonal complement of the harmonic Dirichlet space of the unit disk.

Commuting Dual Toeplitz Operators on the Harmonic Dirichlet Space

In this paper, we characterize the symbols for （semi-）commuting dual Toeplitz operators on the orthogonal complement of the harmonic Dirichlet space. We show that for φ ,ψ ∈ W1, ∞, SφSψ = SψSφ on （Dh）⊥ if and only if φ and ψ satisfy one of the following conditions： （1） Both φ and ψ are harmonic functions; （2） There exist complex constants α and β, not both O, such that φ = αφ＋β.

Dual Toeplitz Operators on the Orthogonal Complement of the Harmonic Bergman Space

In this paper,we characterize that the boundedness,compactness and spectral structure of dual Toeplitz operators acting on the orthogonal complement of the harmonic Bergman space.This generalizes the corresponding results for dual Toeplitz operators on the orthogonal complement of the Bergman space due to Stroethoff and Zheng’s paper[Trans.Amer.Math.Soc.,354,2495–2520(2002)].

Hyponormal Dual Toeplitz Operators on the Orthogonal Complement of the Harmonic Bergman Space

In this paper,we mainly study the hyponormality of dual Toeplitz operators on the orthogonal complement of the harmonic Bergman space.First we show that the dual Toeplitz operator with the bounded symbol is hyponormal if and only if it is normal.Then we obtain a necessary and sufficient condition for the dual Toeplitz operator S_(φ) with the symbol φ(z)=az^(n1zm1)+bz^(n2zm2)(n1,n2,m1,m2∈N and a,b∈C)to be hyponormal.Finally,we show that the rank of the commutator of two dual Toeplitz operators must be an even number if the commutator has a finite rank.

Essentially Commuting Dual Truncated Toeplitz Operators

In this paper,the authors completely characterize when two dual truncated Toeplitz operators ar e essentially commuting and when the semicommutator of two dual truncated Toeplitz operators is compact.Their main idea is to study dual truncated Toeplitz operators via Hankel operators,Toeplitz operators and function algebras.

Dual Toeplitz Algebra on the Polydisk

In this paper,we prove the dual Toeplitz algebra I（C（D^-n））contains the ideal к of compact operators as its semicommutator ideal,and study its algebraic structure.We also get some results about spectrum of dual Toeplitz operators.

Operators on the orthogonal complement of the Dirichlet space (Ⅱ)

In this paper we first prove that a dual Hankel operator Rφ on the orthogonal complement of the Dirichlet space is compact for φ ∈ W^1,∞（D）, and then that a semicommutator of two Toeplitz operators on the Dirichlet space or two dual Toeplitz operators on the orthogonal complement of the Dirichlet space in Sobolev space is compact. We also prove that a dual Hankel operator Re with φ ∈ W^1,∞（D） is of finite rank if and only if Be is orthogonal to the Dirichlet space for some finite Blaschke product B, and give a sufficient and necessary condition for the semicommutator of two dual Toeplitz operators to be of finite rank.