Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access June 6, 2022

Normality and Quasinormality of Specific Bounded Product of Densely Defined Composition Operators in L2 Spaces

  • Hang Zhou EMAIL logo
From the journal Concrete Operators


Let (X, 𝒜, μ) be a σ−finite measure space. A transformation ϕ : XX is non-singular if μϕ−1 is absolutely continuous with respect with μ. For this non-singular transformation, the composition operator Cϕ: 𝒟(Cϕ) → L2(μ) is defined by Cϕf = fϕ, f ∈ 𝒟(Cϕ).

For a fixed positive integer n ≥ 2, basic properties of product Cϕn · · · Cϕ1 in L2(μ) are presented in Section 2, including the boundedness and adjoint. Under the assistance of these properties, normality and quasinormality of specific bounded Cϕn · · · Cϕ1 in L2(μ) are characterized in Section 3 and 4 respectively, where Cϕ1, Cϕ2, · · ·, Cϕn are all densely defined.


[1] R.B. Ash, C. A. Doléans-Dade, Probability and Measure Theory, Harcourt/Academic Press, Burlington, 2000.Search in Google Scholar

[2] P. Budzyński, Z. J. Jabłoński, I. B. Jung and J. Stochel, On unbounded composition operators in L2-spaces, Annali di Matematica, 193 (2014), 663–688.10.1007/s10231-012-0296-4Search in Google Scholar

[3] P. Budzyński, Z. Jabłoński, I. B. Jung and J. Stochel, Unbounded weighted composition operators in L2-Spaces, Lecture Notes in Mathematics 2209, Springer, Switzerland, 2018.10.1007/978-3-319-74039-3Search in Google Scholar

[4] D. L. Cohn, Measure Theory, Second Edition, Springer Science+Business Media, 2013.10.1007/978-1-4614-6956-8Search in Google Scholar

[5] C. C. Cowen and B. D. Maccluer, Composition operators on spaces of analytic functions, CRC Press, 1995.Search in Google Scholar

[6] P. R. Halmos, A Hilbert Space Problem Book, Second Edition, Springer-Verlag, New York, 1974, 1982.10.1007/978-1-4615-9976-0Search in Google Scholar

[7] D. Harrington, R. Whitley, Seminormal composition operators, J. Oper Theory, 11 (1984), 125–135.Search in Google Scholar

[8] B. O. Kppoman, Hamiltonian systems and transformations in Hilbet space, Proc. Natl. Acad. Sci. USA, 17 (1931), 315–318.10.1073/pnas.17.5.315Search in Google Scholar PubMed PubMed Central

[9] J. Neveu, Bases mathématiques du caldul des probabilités (Franch), Éditeurs, Paris, 1970.Search in Google Scholar

[10] E. Nordgren, Composition operators, Canad. J. Math., 20 (1968), 442–449.10.4153/CJM-1968-040-4Search in Google Scholar

[11] E. Nordgren, Composition operators on Hilbert spaces, Lecture Notes in Mathematics, 693 (1978), 37–63.10.1007/BFb0064659Search in Google Scholar

[12] W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1987.Search in Google Scholar

[13] M. M. Rao, Conditional Measures and Applications. Pure and Applied Mathematics, Champan and Hall/CRC, Boca Raton, 2005.Search in Google Scholar

[14] J. Shapiro, The essential norm of a composition operator, Ann. of Math. (2), 125 (1987) no.2, 375–404.Search in Google Scholar

[15] J. Weidmann, Linear Operators in Hilbert Spaces, Springer-Verlag, New York, 1980.10.1007/978-1-4612-6027-1Search in Google Scholar

[16] R. Whitley, Normal and quasinormal composition operators, Proc. Amer. Math. Soc., 70 (1978), no. 2, 114–118.Search in Google Scholar

Received: 2021-10-06
Accepted: 2022-04-26
Published Online: 2022-06-06

© 2022 Hang Zhou, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 29.9.2023 from
Scroll to top button