Abstract:
We consider normal unbounded operators acting in a real Hilbert space. The standard approach to solving spectral problems related with such operators is to apply the complexification, which is a passage to a complex space. At that, usually, the final results are to be decomplexified, that is, the reverse passage is needed. However, the decomplexification often turns out to be nontrivial.
The aim of the present paper is to extend the classical results of the spectral theory for the case of normal operators acting in a real Hilbert space. We provide two real versions of the spectral theorem for such operators.
We construct the functional calculus generated by the real spectral decomposition of a normal operator. We provide examples of using the obtained functional calculus for representing the exponent of a normal operator.
Keywords:
unbounded normal operator, real Hilbert space, complexification, spectral theorem, functional calculus.
\Bibitem{Ore17}
\by M.~N.~Oreshina
\paper Spectral decomposition of normal operator in real Hilbert space
\jour Ufa Math. J.
\yr 2017
\vol 9
\issue 4
\pages 85--96
\mathnet{http://mi.mathnet.ru/eng/ufa408}
\crossref{https://doi.org/10.13108/2017-9-4-85}
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Linking options:
https://www.mathnet.ru/eng/ufa408
https://doi.org/10.13108/2017-9-4-85
https://www.mathnet.ru/eng/ufa/v9/i4/p87
This publication is cited in the following 4 articles:
Zhi-jie Jiang, “Complex symmetric difference of the weighted composition operators on weighted Bergman space of the half-plane”, MATH, 9:3 (2024), 7253
Roman Gielerak, “Renormalized Von Neumann entropy with application to entanglement in genuine infinite dimensional systems”, Quantum Inf Process, 22:8 (2023)
Maria Oreshina, 2020 2nd International Conference on Control Systems, Mathematical Modeling, Automation and Energy Efficiency (SUMMA), 2020, 125
J. Li, X. Yan, M. Li, M. Meng, X. Yan, “A method of fpga-based extraction of high-precision time-difference information and implementation of its hardware circuit”, Sensors, 19:23 (2019), 5067