Essential norm of generalized Hilbert matrix from Bloch type spaces to BMOA and Bloch space

Songxiao Li; Jizhen Zhou; Songxiao Li; Jizhen Zhou

doi:10.3934/math.2021198

AIMS Mathematics

2021, Volume 6, Issue 4: 3305-3318. doi: 10.3934/math.2021198

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Research article

Essential norm of generalized Hilbert matrix from Bloch type spaces to BMOA and Bloch space

Songxiao Li ¹,
Jizhen Zhou ^{2
,
,}

1.
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, 610054, Chengdu, Sichuan, P. R. China
2.
School of Sciences, Anhui University of Science and Technology, 232001, Huainan, Anhui, P. R. China

Received: 09 November 2020 Accepted: 06 January 2021 Published: 18 January 2021
MSC : 47B38, 30H30

Let $ \mu $ be a positive Borel measure on the interval $ [0, 1) $. The Hankel matrix $ {\mathcal H}_\mu = (\mu_{n+k})_{n, k\geq 0} $ with entries $ \mu_{n, k} = \mu_{n+k} $ induces the operator $ {\mathcal H}_\mu(f)(z) = \sum\limits_{n = 0}^\infty\left(\sum\limits_{k = 0}^\infty\mu_{n,k}a_k\right)z^n $ on the space of all analytic functions $ f(z) = \sum^\infty_{n = 0}a_nz^n $ in the unit disk $ {\mathbb{D}} $. In this paper, we characterize the boundedness and compactness of $ {\mathcal H}_\mu $ from Bloch type spaces to the BMOA and the Bloch space. Moreover we obtain the essential norm of $ {\mathcal H}_\mu $ from $ {\mathcal{B}}^\alpha $ to $ {\mathcal{B}} $ and BMOA.
- Bloch type space,
- BMOA space,
- Carleson measure,
- Hilbert operator,
- essential norm
Citation: Songxiao Li, Jizhen Zhou. Essential norm of generalized Hilbert matrix from Bloch type spaces to BMOA and Bloch space[J]. AIMS Mathematics, 2021, 6(4): 3305-3318. doi: 10.3934/math.2021198

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Abstract

Let $ \mu $ be a positive Borel measure on the interval $ [0, 1) $. The Hankel matrix $ {\mathcal H}_\mu = (\mu_{n+k})_{n, k\geq 0} $ with entries $ \mu_{n, k} = \mu_{n+k} $ induces the operator $ {\mathcal H}_\mu(f)(z) = \sum\limits_{n = 0}^\infty\left(\sum\limits_{k = 0}^\infty\mu_{n,k}a_k\right)z^n $ on the space of all analytic functions $ f(z) = \sum^\infty_{n = 0}a_nz^n $ in the unit disk $ {\mathbb{D}} $. In this paper, we characterize the boundedness and compactness of $ {\mathcal H}_\mu $ from Bloch type spaces to the BMOA and the Bloch space. Moreover we obtain the essential norm of $ {\mathcal H}_\mu $ from $ {\mathcal{B}}^\alpha $ to $ {\mathcal{B}} $ and BMOA.

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