Composition operators on Hardy-Sobolev spaces with bounded reproducing kernels

Li He; Li He

doi:10.3934/math.2023142

AIMS Mathematics

2023, Volume 8, Issue 2: 2708-2719. doi: 10.3934/math.2023142

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Research article

Composition operators on Hardy-Sobolev spaces with bounded reproducing kernels

Li He ^,

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

Received: 17 July 2022 Revised: 19 September 2022 Accepted: 10 October 2022 Published: 09 November 2022
MSC : 47B33, 47A53

For any real $ \beta $ let $ H^2_\beta $ be the Hardy-Sobolev space on the unit disc $ {\mathbb D} $. $ H^2_\beta $ is a reproducing kernel Hilbert space and its reproducing kernel is bounded when $ \beta > 1/2 $. In this paper, we prove that $ C_{\varphi } $ has dense range in $ H_{\beta }^{2} $ if and only if the polynomials are dense in a certain Dirichlet space of the domain $ \varphi({\mathbb D}) $ for $ 1/2 < \beta < 1 $. It follows that if the range of $ C_{\varphi } $ is dense in $ H_{\beta }^{2} $, then $ \varphi $ is a weak-star generator of $ H^{\infty} $, although the conclusion is false for the classical Dirichlet space $ \mathfrak{D} $. Moreover, we study the relation between the density of the range of $ C_{\varphi } $ and the cyclic vector of the multiplier $ M_{\varphi}^{\beta}. $
- Hardy-Sobolev space,
- composition operator,
- reproducing kernel,
- automorphism
Citation: Li He. Composition operators on Hardy-Sobolev spaces with bounded reproducing kernels[J]. AIMS Mathematics, 2023, 8(2): 2708-2719. doi: 10.3934/math.2023142

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Abstract

For any real $ \beta $ let $ H^2_\beta $ be the Hardy-Sobolev space on the unit disc $ {\mathbb D} $. $ H^2_\beta $ is a reproducing kernel Hilbert space and its reproducing kernel is bounded when $ \beta > 1/2 $. In this paper, we prove that $ C_{\varphi } $ has dense range in $ H_{\beta }^{2} $ if and only if the polynomials are dense in a certain Dirichlet space of the domain $ \varphi({\mathbb D}) $ for $ 1/2 < \beta < 1 $. It follows that if the range of $ C_{\varphi } $ is dense in $ H_{\beta }^{2} $, then $ \varphi $ is a weak-star generator of $ H^{\infty} $, although the conclusion is false for the classical Dirichlet space $ \mathfrak{D} $. Moreover, we study the relation between the density of the range of $ C_{\varphi } $ and the cyclic vector of the multiplier $ M_{\varphi}^{\beta}. $

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