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)
  • ORCID iD 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
  • ORCID iD 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.
  • Received: 2020-01-10.
  • Revised: 2020-05-13.
  • Accepted: 2020-05-29.
  • Published online: 2020-07-09.
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Cite this article as:
Ching-on Lo, Anthony Wai-keung Loh, Hilbert-Schmidtness of weighted composition operators and their differences on Hardy spaces, Opuscula Math. 40, no. 4 (2020), 495-507, https://doi.org/10.7494/OpMath.2020.40.4.495

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