Multiplication Operators on the Bergman Space

Guo, Kun; Huang, Hansong

doi:10.1007/978-3-662-46845-6

Cited by 37 publications

(29 citation statements)

References 58 publications

(202 reference statements)

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“…The reducing subspaces of some analytic Toeplitz operators on the Bergman space of the unit disk were studied in [12], and more general weighted shifts were discussed in [11]. The paper by Zhu [12] was also inspirational in the recent development of reducing subspaces of analytic Toeplitz operators with Blaschke product symbols on the Bergman space [2]. The reducing subspaces of some analytic Toeplitz operators on the Bergman space of the bidisk were characterized in [6] and [9].…”

Section: Introductionmentioning

confidence: 99%

Reducing Subspaces of Weighted Shifts With Operator Weights

Gu¹

2016

Bulletin of the Korean Mathematical Society

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Section: Introductionmentioning

confidence: 99%

Reducing Subspaces of Weighted Shifts With Operator Weights

Gu¹

2016

Bulletin of the Korean Mathematical Society

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“…On the other hand, there is a nice description of reducing subspaces of powers of weighted shifts with scalar weights [21]. This paper and its predecessor [23], where reducing subspaces of some analytic Toeplitz operators on the Bergman space of the unit disk were studied, have also been inspirational in the last fifteen years for establishing C. Gu structure of reducing subspaces of Toeplitz operators with Blaschke product symbols on the Bergman space of the unit disk; see a recent monograph [6] and extensive references therein. The structure of the reducing subspace lattice for unweighted unilateral shifts was described in [9] and [16].…”

Section: Introductionmentioning

confidence: 93%

Common reducing subspaces of several weighted shifts with operator weights

Gu¹

2018

J. Math. Soc. Japan

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We characterize common reducing subspaces of several weighted shifts with operator weights. As applications, we study the common reducing subspaces of the multiplication operators by powers of coordinate functions on Hilbert spaces of holomorphic functions in several variables. The identification of reducing subspaces also leads to structure theorems for the commutants of von Neumann algebras generated by these multiplication operators. This general approach applies to weighted Hardy spaces, weighted Bergman spaces, Drury-Arveson spaces and Dirichlet spaces of the unit ball or polydisk uniformly.

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“…While the invariant subspaces of M θ are the same with respect to equivalent norms, the reducing subspaces of M θ could be different with respect to equivalent norms. Starting with the paper [32] in 2000 where the reducing subspaces of M θ for a Blaschke product of order 2 on the Bergman space were characterized, there have been intensive research activities to understand the reducing subspaces of M θ on the Bergman space for a general Blaschke product, see the recent book [16]. A recent paper [23] discusses the reducing subspaces of M θ on the Dirichlet space.…”

Section: Blaschke Products and Reducing Subspacesmentioning

confidence: 99%

“…Multiplication operators and composition operators on subspaces of H(D) such as the Hardy space, the Bergman space and the Dirichlet space have been studied extensively in literature [2] [10] [12] [16].…”

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confidence: 99%

Composition and multiplication operators on the derivative Hardy space

Luo

2017

Complex Variables and Elliptic Equations

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In this paper we propose a different (and equivalent) norm on S 2 (D) which consists of functions whose derivatives are in the Hardy space of unit disk. The reproducing kernel of S 2 (D) in this norm admits an explicit form, and it is a complete Nevanlinna-Pick kernel. Furthermore, there is a surprising connection of this norm with 3-isometries. We then study composition and multiplication operators on this space. Specifically, we obtain an upper bound for the norm of C ϕ for a class of composition operators. We completely characterize multiplication operators which are m-isometries. As an application of the 3-isometry, we describe the reducing subspaces of M ϕ on S 2 (D) when ϕ is a finite Blaschke product of order 2.

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Multiplication Operators on the Bergman Space

Cited by 37 publications

References 58 publications

Reducing Subspaces of Weighted Shifts With Operator Weights

Reducing Subspaces of Weighted Shifts With Operator Weights

Common reducing subspaces of several weighted shifts with operator weights

Composition and multiplication operators on the derivative Hardy space

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