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Stochastic Processes and Spectral Analysis for Hilbert \(C^*\)-Module-Valued Maps

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Abstract

In this paper, we consider Hilbert \(C^*\)-module-valued random variables and discuss about their expectations, covariance operators and correlation operators by introducing some adjointable operators on Hilbert \(C^*\)-modules. We also study Hilbert \(C^*\)-module-valued stochastic processes instead of Hilbert space-valued processes as a generalization. Using the characterization of \(C^*\)-algebra-valued bimeasures, we prove the equivalence between V-boundedness and harmonizability of Hilbert \(C^*\)-module-valued stochastic processes. Finally, we consider Hilbert \(C^*\)-module operator-valued stochastic processes and construct spectral distributions associated with Hilbert \(C^*\)-module operator-valued stationary stochastic processes.

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Acknowledgments

A part of this research was carried out while the author was visiting the Korean Institute for Advanced Study (KIAS). He is grateful to professors and the staff in the KIAS for their warm hospitality. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT, and future Planning (2014029581).

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Authors and Affiliations

  1. Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, South Korea

    Jaeseong Heo

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  1. Jaeseong Heo
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Correspondence to Jaeseong Heo.

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Communicated by Mohammad Sal Moslehian.

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Heo, J. Stochastic Processes and Spectral Analysis for Hilbert \(C^*\)-Module-Valued Maps. Bull. Malays. Math. Sci. Soc. 41, 191–206 (2018). https://doi.org/10.1007/s40840-015-0270-6

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  • DOI: https://doi.org/10.1007/s40840-015-0270-6

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