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.

Full text (pdf)

1. W. Al-Rawashdeh, S.K. Narayan, Difference of composition operators on Hardy space, J. Math. Inequal. 7 (2013), 427-444.
2. E. Berkson, Composition operators isolated in the uniform operator topology, Proc. Amer. Math. Soc. 81 (1981), 230-232.
3. P.S. Bourdon, Components of linear-fractional composition operators, J. Math. Anal. Appl. 279 (2003), 228-245.
4. D. Buchholz, C. D'Antoni, R. Longo, Nuclear maps and modular structures. I. General properties, J. Funct. Anal. 88 (1990), 233-250.
5. B.R. Choe, T. Hosokawa, H. Koo, Hilbert-Schmidt differences of composition operators on the Bergman space, Math. Z. 269 (2011), 751-775.
6. J.B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 2007.
7. C.C. Cowen, B.D. MacCluer, Composition operators on spaces of analytic functions, CRC Press, Boca Raton, Florida, 1995.
8. P.L. Duren, Theory of $$H^p$$ Spaces, Academic Press, New York, 1970 (Reprinted by Dover, Mineola, New York, 2000).
9. E.A. Gallardo-Gutiérrez, M.J. González, P.J. Nieminen, E. Saksman, On the connected component of compact composition operators on the Hardy space, Adv. Math. 219 (2008), 986-1001.
10. K. Hoffman, Banach spaces of analytic functions, Dover Publications, New York, 1962.
11. P. Lefèvre, D. Li, H. Queffélec, L. Rodríguez-Piazza, Some new properties of composition operators associated with lens maps, Israel J. Math. 195 (2013), 801-824.
12. D. Li, H. Queffélec, L. Rodríguez-Piazza, On approximation numbers of composition operators, J. Approx. Theory 164(4) (2012), 431-459.
13. V. Matache, Weighted composition operators on $$H^2$$ and applications, Complex Anal. Oper. Theory 2 (2008), 169-197.
14. T.Y. Na, Computational methods in engineering boundary value problems, Mathematics in Science and Engineering, vol. 145, Academic Press, 1979.
15. J.H. Shapiro, C. Sundberg, Isolation amongst the composition operators, Pacific J. Math. 145 (1990), 117-152.
• 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.