Polynomial approximation and composition operators

Cao, G. F.; He, Li; Zhu, Kehe

doi:10.1090/proc/14455

Cited by 8 publications

(8 citation statements)

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“…The first author of this paper and his coworkers (K.H. Zhu and L. He) find that the density of the polynomials in D(Ω) is equivalent to the density of the range of the composition operators C ϕ acting on D(D), and obtain the following characterization results in [2], which partly solves the open problem 1.1.…”

Section: Introductionmentioning

confidence: 70%

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Composition Operators on Dirichlet Spaces over the Half-plane

Cao¹,

Li²

2022

Preprint

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As continuation of the study of polynomial approximation and composition operators on Dirichlet spaces of unit disk, which has settled a problem posed by Cima in 1976, the present paper aims to consider the case of the unbounded domains, such as the half-plane. Specifically, we may obtain the rational approximations in the Dirichlet spaces and characterize the composition operators which has dense range on the Dirichlet spaces over the half-plane. Moreover, this paper also considers the relationship between the Dirichlet spaces and Hardy spaces on half-plane.Problem 1.1. Characterize those ϕ in the unit ball of H ∞ for which C ϕ has dense range.Cima indicated that if C ϕ has dense range, then for ψ(z) = rϕ(z), 0 < r < 1, C ψ has dense range. He said that no conjecture on this question. This is an old 2010 Mathematics Subject Classification. 30H25.

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Section: Introductionmentioning

confidence: 70%

“…Theorem A ( [2]) Suppose ϕ : D → D is analytic and Ω = ϕ(D). Then C ϕ is a bounded composition operator on Dirichlet spaces D(D) with dense range if and only if the polynomials are dense in the Bergman space A 2 (Ω) and the self-mapping ϕ is univalent.…”

Section: Introductionmentioning

confidence: 99%

Composition Operators on Dirichlet Spaces over the Half-plane

Cao¹,

Li²

2022

Preprint

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“…Bourdon and Roan studied the problem for the Hardy space (see [2,22]) and Cima raised the problem for the Dirichlet space in [10]. In [7], we settled Cima's problem completely: Theorem 2. Suppose ϕ : D → D is analytic, non-constant, and G = ϕ(D).…”

Section: Proposition 1 ([6]mentioning

confidence: 91%

“…if and only if the polynomials are dense in A 2 (ϕ(D), (1 − |ϕ −1 | 2 )dA) (see [2]), and R(C ϕ ) is dense in D if and only if the polynomials are dense in A 2 (ϕ(D)) (see [7]).…”

Section: Polynomial Approximation and Composition Operatorsmentioning

confidence: 99%

Composition operators on Hardy-Sobolev spaces with bounded reproducing kernels

Cao¹,

He²

2021

Preprint

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For any real β let H 2 β be the Hardy-Sobolev space on the unit disk D. H 2 β is a reproducing kernel Hilbert space and its reproducing kernel is bounded when β > 1/2. In this paper, we study composition operators C ϕ on H 2 β for 1/2 < β < 1. Our main result is that, for a non-constant analytic function ϕ : D → D, the operator C ϕ has dense range in H 2 β if and only if the polynomials are dense in a certain Dirichlet space of the domain ϕ(D). It follows that if the range of C ϕ is dense in H 2 β , then ϕ is a weak-star generator of H ∞ . Note that this conclusion is false for the classical Dirichlet space D. We also characterize Fredholm composition operators on H 2 β . 2010 Mathematics Subject Classification. 47B33, 47A53.

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“…Finally, in the last section we discuss the similar problem in general domains (see §3). We close this section by mentioning that polynomial approximation is closely connected to the invariant subspaces of the space in question (see [1] and [2]); as well as to the behavior of composition operators defined on the underlying domain, see for instance [3]. This issue also pops up in the decomposition of functions in the weighted Bergman spaces; see a recent paper of Korhonen and Rättyä where the authors assume that the weight is chosen in such a way that the polynomials are dense in the weighted Bergman space A p w ([10], Theorem 4).…”

Section: Introductionmentioning

confidence: 99%