The rank of Hankel operators on harmonic Bergman spaces

Let f and g be bounded functions, and let T f and T g be Toeplitz operators on A 2 2 D . We show that if the product T f T g equals zero and one of f and g is a radial function satisfying a Mellin transform condition, then the other function must be zero.

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“…where gðrÞ is a radial function (see [14]). For any function f ∈ L 2 ðD, dAÞ, it has the polar decomposition, i.e.,…”

Section: Products Of Two Toeplitz Operatorsmentioning

confidence: 99%

Products of Toeplitz Operators on the 2-Analytic Bergman Space

Zhang

Yang

2021

Journal of Function Spaces

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“…(α) z n (z k ). It is shown in [12] that if Ψ and Ω are two functions in L 2 a (dA α ) such that Ψ(0) = 0 = Ω(0) then the operators B is linearly independent for all p > 0. If g ∈ L 2 h (dA α ) and g = Ψ + Ω, where Ψ and Ω are from L 2 a (dA α ) then…”

Section: •Bmentioning

confidence: 99%

Fredholm Toeplitz Operators on the Weighted Bergman Spaces

Das¹,

Roy²

2021

AnnalsARSciMath

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In this paper we have shown that if φ ∈ (L2h(dAα))⊥∩ L∞(D) andRangeT(α)φis closed, then the Toeplitz operator T(α)φ∈ LL2a(dAα)is a Fredholm operator of index zero and T(α)φis not of ﬁnite rank.Several applications of the result were also obtained. We further showthat if φ ∈ L∞Mn(D) is such that Tφis Fredholm and of index zero inLL2,Cna(dAα)then there exists ψ ∈ En×n= E ⊗Mnsuch that Tφ+δψis invertible for all suﬃciently small nonzero δ. Here E is a total sub-space of L∞(D) and Mnis the set of all n × n matrices with complexentries.

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“…In this subsection, commuting Toeplitz operators with bounded quasihomogeneous symbols on the Dirichlet-type space of the unit ball are discussed. The definition of the quasihomogeneous function on the unit disk has been given in [17,26], and the definition on the unit ball has been given in [22]. Definition 18.…”

Section: Commuting Toeplitz Operators With Quasihomogeneousmentioning

confidence: 99%

Toeplitz Operators on Dirichlet-Type Space of Unit Ball

Jin

Wang

Cao

2014

Abstract and Applied Analysis

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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.

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The rank of Hankel operators on harmonic Bergman spaces

Abstract: Abstract. We show that on the harmonic Bergman spaces, the Hankel operators with nonconstant harmonic symbol cannot be of finite rank.

Cited by 10 publications

References 4 publications

Products of Toeplitz Operators on the 2-Analytic Bergman Space

Products of Toeplitz Operators on the 2-Analytic Bergman Space

Fredholm Toeplitz Operators on the Weighted Bergman Spaces

Toeplitz Operators on Dirichlet-Type Space of Unit Ball

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