Multiplication Operators Defined by a Class of Polynomials on $${L_a^2(\mathbb{D}^2)}$$ L a 2 ( D 2 )

Dan, Hui; Huang, Hansong

doi:10.1007/s00020-014-2176-3

Cited by 15 publications

(6 citation statements)

References 26 publications

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“…Then H 2 (ω, δ) is a graded S-module, and Theorem 4.5 reveals that [1] S is a minimal reducing subspace. In this section, we mainly give a concise summary of the results in [12], [25]. There are two primary questions:…”

Section: Z+w On H 2 (ω δ)mentioning

confidence: 99%

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The graded structure induced by operators on a Hilbert space

Guo¹,

Wang²

2018

J. Math. Soc. Japan

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In this paper we define a graded structure induced by operators on a Hilbert space. Then we introduce several concepts which are related to the graded structure and examine some of their basic properties. A theory concerning minimal property and unitary equivalence is then developed. It allows us to obtain a complete description of V * (M z k ) on any H 2 (ω). It also helps us to find that a multiplication operator induced by a quasi-homogeneous polynomial must have a minimal reducing subspace. After a brief review of multiplication operator M z+w on H 2 (ω, δ), we prove that the Toeplitz operator T z+w on H 2 (D 2 ), the Hardy space over the bidisk, is irreducible.

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Section: Z+w On H 2 (ω δ)mentioning

confidence: 99%

“…Intuitively, the graded structures of Hilbert spaces that Hilbert polynomials can be defined on should attract some attentions. Lately, motivated by [12], [25], [32], we find that graded structure can be very useful in operator theory.…”

Section: Introductionmentioning

confidence: 99%

The graded structure induced by operators on a Hilbert space

Guo¹,

Wang²

2018

J. Math. Soc. Japan

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“…If is a monomial, the reducing subspaces of are characterized in [14][15][16][17]. If = 1 + 2 , Dan and Huang [18] described the minimal reducing subspaces of and the commutant algebra { , * } .…”

Section: Introductionmentioning

confidence: 99%

Reducing Subspaces of Some Multiplication Operators on the Bergman Space over Polydisk

Shi

Zhou

2015

Abstract and Applied Analysis

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We consider the reducing subspaces of on 2 (D ), where ≥ 3, = 1 1 ⋅ ⋅ ⋅ , and ̸ = for ̸ = . We prove that each reducing subspace of is a direct sum of some minimal reducing subspaces. We also characterize the minimal reducing subspaces in the cases that = 0 and ∈ (−1, +∞) \ Q, respectively. Finally, we give a complete description of minimal reducing subspaces of on 2 (D 3 ) with > −1.

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“…For p = αz k + βw l , the minimal reducing subspaces of T p on A 2 (D 2 ) and the commutant algebra V * (p) = {T p , T * p } ′ was described in [1,15]. In this paper, we mainly consider the reducing subspaces for the Toeplitz operator…”

Section: Introductionmentioning

confidence: 99%

Reducing Subspaces for a Class of Toeplitz Operators on the Bergman Space of the Bidisk

Albaseer¹,

Lu²,

Shi³

2015

Bulletin of the Korean Mathematical Society

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Multiplication Operators Defined by a Class of Polynomials on $${L_a^2(\mathbb{D}^2)}$$ L a 2 ( D 2 )

Abstract: In this paper, we consider those multiplication operators M p on L 2 a

Cited by 15 publications

References 26 publications

The graded structure induced by operators on a Hilbert space

The graded structure induced by operators on a Hilbert space

Reducing Subspaces of Some Multiplication Operators on the Bergman Space over Polydisk

Reducing Subspaces for a Class of Toeplitz Operators on the Bergman Space of the Bidisk

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