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Commuter of operators in a Hilbert space

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  • Boudou, Alain
  • Viguier-Pla, Sylvie

Abstract

Commutativity between two stationary functions is a notion that generalizes the concept of stationary correlation. Two stationary functions commute if and only if their associated spectral measures and unitary operators commute. When the operators do not commute, we define the concept of commuter for two operators, and then derive it for two projectors, for a projector and a unitary operator, and for two unitary operators. We establish relations between these commuters and some other tools related to proximity between processes. Finally, we propose a method which retrieves the commuter for two stationary series of finite spectrum, and we study some convergence properties.

Suggested Citation

  • Boudou, Alain & Viguier-Pla, Sylvie, 2019. "Commuter of operators in a Hilbert space," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 244-262.
    • Handle: RePEc:eee:jmvana:v:170:y:2019:i:c:p:244-262
    DOI: 10.1016/j.jmva.2018.06.011
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    References listed on IDEAS

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    1. Boudou, Alain & Romain, Yves, 2002. "On spectral and random measures associated to discrete and continuous-time processes," Statistics & Probability Letters, Elsevier, vol. 59(2), pages 145-157, September.
    2. Ferraty, Frederic & Vieu, Philippe & Viguier-Pla, Sylvie, 2007. "Factor-based comparison of groups of curves," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4903-4910, June.
    3. Boudou, Alain & Viguier-Pla, Sylvie, 2016. "Gap between orthogonal projectors—Application to stationary processes," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 282-300.
    4. Kneip, Alois & Benko, Michal, 2005. "Common functional component modelling," SFB 649 Discussion Papers 2005-016, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    5. Boudou, Alain, 2006. "Approximation of the principal components analysis of a stationary function," Statistics & Probability Letters, Elsevier, vol. 76(6), pages 571-578, March.
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    Cited by:

    1. Aneiros, Germán & Cao, Ricardo & Fraiman, Ricardo & Genest, Christian & Vieu, Philippe, 2019. "Recent advances in functional data analysis and high-dimensional statistics," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 3-9.

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