Geodesic
Spectral decomposition of normal operator in real Hilbert space
Ufa mathematical journal, Tome 9 (2017) no. 4, pp. 85-96 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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, spectral theorem, functional calculus.
Mots-clés : complexification 
@article{UFA_2017_9_4_a8,
     author = {M. N. Oreshina},
     title = {Spectral decomposition of normal operator in real {Hilbert} space},
     journal = {Ufa mathematical journal},
     pages = {85--96},
     year = {2017},
     volume = {9},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2017_9_4_a8/}
}
TY  - JOUR
AU  - M. N. Oreshina
TI  - Spectral decomposition of normal operator in real Hilbert space
JO  - Ufa mathematical journal
PY  - 2017
SP  - 85
EP  - 96
VL  - 9
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/UFA_2017_9_4_a8/
LA  - en
ID  - UFA_2017_9_4_a8
ER  - 
%0 Journal Article
%A M. N. Oreshina
%T Spectral decomposition of normal operator in real Hilbert space
%J Ufa mathematical journal
%D 2017
%P 85-96
%V 9
%N 4
%U http://geodesic.mathdoc.fr/item/UFA_2017_9_4_a8/
%G en
%F UFA_2017_9_4_a8
M. N. Oreshina. Spectral decomposition of normal operator in real Hilbert space. Ufa mathematical journal, Tome 9 (2017) no. 4, pp. 85-96. http://geodesic.mathdoc.fr/item/UFA_2017_9_4_a8/

[1] Moren K., Metody gilbertova prostranstva, Mir, M., 1965, 572 pp. | MR

[2] Naimark M. A., Normirovannye koltsa, Nauka, M., 1968, 664 pp. | MR

[3] Rudin U., Funktsionalnyi analiz, Mir, M., 1975, 444 pp. | MR

[4] Khille E., Fillips R., Funktsionalnyi analiz i polugruppy, Inostrannaya literatura, M., 1962, 830 pp. | MR

[5] Danford N., Shvarts Dzh.T., Lineinye operatory. Spektralnaya teoriya, Mir, M., 1966, 1065 pp.

[6] Golub Dzh., Van Loun Ch., Matrichnye vychisleniya, Mir, M., 1999, 548 pp.

[7] Demmel Dzh., Vychislitelnaya lineinaya algebra. Teoriya i prilozheniya, Mir, M., 2001, 430 pp.

[8] A. C. Antoulas, Approximation of large-scale dynamical systems, SIAM, Philadelphia, 2005, 479 pp. | MR | Zbl

[9] I. P. Gavrilyuk, V. L. Makarov, “Exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in Banach spaces”, SIAM Journal on Numerical Analysis, 43:5 (2005), 2144–2171 | DOI | MR | Zbl

[10] L. Lopez, V. Simoncini, “Analysis of projection methods for rational function approximation to the matrix exponential”, SIAM Journal on Numerical Analysis, 44:2 (2006), 613–635 | DOI | MR | Zbl

[11] Kurbatov V. G., Oreshina M. N., “O nakhozhdenii priblizhennogo resheniya lineinogo differentsialnogo uravneniya vtorogo poryadka”, Vestnik VGU. Seriya: fizika, matematika, 2003, no. 2, 173–188 | Zbl

[12] V.G. Kurbatov, I.V. Kurbatova, “Krylov subspace methods of approximate solving differential equations from the point of view of functional calculus”, Eurasian Mathematical Journal, 3:4 (2012), 53–80 | MR | Zbl

[13] V.G. Kurbatov, M.N. Oreshina, “Interconnect macromodelling and approximation of matrix exponent”, Analog Integrated Circuits and Signal Processing, 40:1 (2004), 5–19 | DOI

[14] Oreshina M. N., “Priblizhennoe reshenie parabolicheskogo uravneniya s ispolzovaniem ratsionalnoi approksimatsii operatornoi eksponenty”, Differentsialnye uravneniya, 53:3 (2017), 407–417 | DOI | MR | Zbl

[15] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, Nauka 1984, M., 752 pp. | MR

[16] Baskakov A. G., Zagorskii A. S., “K spektralnoi teorii lineinykh otnoshenii”, Matem. zametki, 81:1 (2007), 17–31 | DOI | MR | Zbl

[17] Glazman I. M., Lyubich Yu. I., Konechnomernyi lineinyi analiz, Nauka, M., 1969, 476 pp. | MR

[18] Pechkurov A. V., “Kompleksifikatsiya uporyadochennoi pary lineinykh operatorov”, Vestnik VGU. Seriya: fizika, matematika, 2007, no. 2, 143–147

[19] Oreshina M. N., “Spektralnaya teorema dlya samosopryazhennogo operatora v deistvitelnom gilbertovom prostranstve”, Vestnik VGU. Seriya: fizika, matematika, 2015, no. 3, 120–133 | Zbl

[20] Khalmosh P., Teoriya mery, Inostrannaya literatura, M., 1953, 292 pp. | MR

[21] Burbaki N., Integrirovanie. Mery na lokalno kompaktnykh prostranstvakh. Prodolzhenie mery. Integrirovanie mer. Mery na otdelimykh prostranstvakh, Nauka, M., 1977, 601 pp.

[22] Krein S. G., Lineinye differentsialnye uravneniya v banakhovom prostranstve, Nauka, M., 1967, 464 pp. | MR