Opuscula Math. 40, no. 4 (2020), 495-507
https://doi.org/10.7494/OpMath.2020.40.4.495

Opuscula Mathematica

Hilbert-Schmidtness of weighted composition operators and their differences on Hardy spaces

Ching-on Lo
Anthony Wai-keung Loh

Abstract. Let $$u$$ and $$\varphi$$ be two analytic functions on the unit disk $$\mathbb{D}$$ such that $$\varphi(\mathbb{D}) \subset \mathbb{D}$$. A weighted composition operator $$uC_{\varphi}$$ induced by $$u$$ and $$\varphi$$ is defined on $$H^2$$, the Hardy space of $$\mathbb{D}$$, by $$uC_{\varphi}f := u \cdot f \circ \varphi$$ for every $$f$$ in $$H^2$$. We obtain sufficient conditions for Hilbert-Schmidtness of $$uC_{\varphi}$$ on $$H^2$$ in terms of function-theoretic properties of $$u$$ and $$\varphi$$. Moreover, we characterize Hilbert-Schmidt difference of two weighted composition operators on $$H^2$$.

Keywords: weighted composition operators, Hardy spaces, compact operators, Hilbert-Schmidt operators.

Mathematics Subject Classification: 47B33, 30H10.

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• Ching-on Lo (corresponding author)
• https://orcid.org/0000-0003-2735-8726
• Division of Science, Engineering and Health Studies, College of Professional and Continuing Education, The Hong Kong Polytechnic University
• Anthony Wai-keung Loh
• https://orcid.org/0000-0002-2759-3198
• Division of Science, Engineering and Health Studies, College of Professional and Continuing Education, The Hong Kong Polytechnic University
• Communicated by P.A. Cojuhari.
• Revised: 2020-05-13.
• Accepted: 2020-05-29.
• Published online: 2020-07-09.