Invertible weighted composition operators preserve frames on Dirichlet type spaces

Ruishen Qian; Xiangling Zhu; Ruishen Qian; Xiangling Zhu

doi:10.3934/math.2020273

AIMS Mathematics

2020, Volume 5, Issue 5: 4285-4296. doi: 10.3934/math.2020273

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Research article

Invertible weighted composition operators preserve frames on Dirichlet type spaces

Ruishen Qian ¹,
Xiangling Zhu ^{2
,
,}

1.
School of Mathematics and Statistics, Lingnan Normal University, 524048, Zhanjiang, Guangdong, P. R. China
2.
University of Electronic Science and Technology of China, Zhongshan Institute, 528402, Zhongshan, Guangdong, P. R. China

Received: 11 February 2020 Accepted: 30 April 2020 Published: 08 May 2020
MSC : 30H99, 47B33

Some characterizations for weighted composition operators to be invertible on Dirichlet type spaces $\mathfrak{D}_{\rho}$ are given in this paper when $\rho$ is finite lower type greater than $0$ and upper type less than $1$. In particular, the equivalence between invertible and preserve frames is established. Moreover, weighted composition operators that preserve tight frames and normalized tight frames on the Dirichlet type space $\mathfrak{D}_{\alpha}$ $(0 < \alpha < 1)$ are also investigated.
- Dirichlet type space,
- weighted composition operator,
- frame,
- bounded below,
- multiplier
Citation: Ruishen Qian, Xiangling Zhu. Invertible weighted composition operators preserve frames on Dirichlet type spaces[J]. AIMS Mathematics, 2020, 5(5): 4285-4296. doi: 10.3934/math.2020273

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Abstract

Some characterizations for weighted composition operators to be invertible on Dirichlet type spaces $\mathfrak{D}_{\rho}$ are given in this paper when $\rho$ is finite lower type greater than $0$ and upper type less than $1$. In particular, the equivalence between invertible and preserve frames is established. Moreover, weighted composition operators that preserve tight frames and normalized tight frames on the Dirichlet type space $\mathfrak{D}_{\alpha}$ $(0 < \alpha < 1)$ are also investigated.

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