Spectral properties of linear operators on Hilbert quaternionic bimodules (in Russian)

I.I.Karpenko; D.L.Tyshkevich

In the paper, besides classical systems of the axioms accepted in the theory of bimodules over associative rings and in the theory of Hilbert quaternionic modules, the additional axioms taking place in the most common realization of Hilbert modules are offered. Independence of the given axioms of existing systems and also their importance for effective work with Hilbert bimodules are shown. We offer a classification of spectrum of a linear operator over Hilbert quaternionic bimodule which is completely coordinated to classification of spectrum of a complex linear operator. At the same time this classification gives some new properties of the spectrum.

1. Березин А.В., Курочкин Ю.А., Толкачев Е.А. Кватернионы в релятивистской физике.--М.: Едиториал УРСС.~--~2003.~--~200 с.

2. De Leo S., Ducati G. Quaternionic differential operators// arXiv:math-ph/0005023v3.~--~2002.~--~P.1--25.

3. De Leo S., Rotelly P. Translation between Quaternion and Complex Quantum Mechanics// arXiv:hep-th/9401009 v1.~--~1994.~--~P.1--15.

4. Курош А.Г. Лекции по общей алгебре.-- М.: Наука.~--~1973.~--~399 с.

5. Фейс К. Алгебра: кольца, поля, модули и категории, т.1,2. -- М.: Мир.~--~1977. ~--~688,463 с.

6. De Leo S., Scolarici G. Right eigenvalue equation in quaternionic quantum mechanics// Journal Phisics. A33.~--~2000.~--~P.2971--2995.

7. De Leo S., Scolarici G., Solombino L. Quaternionic eigenvalue problem// J.Math. Phys. Bd.43.~--~2002.~--~V.11.~--~P.5815--5829.

8. Sharma C.S., Coulson T.J. Spectral theory for unitary operators on a quaternionic Hilbert space// J.Math.Phys.~--~1987. -- 28(9).~--~P.1941--1945.

9. Куратовский К. Теория множеств.--М.: Мир. ~--~1970.~--~460 с.

10. Abdel-Khalek Kh. Quaternion analysis// arXiv:hep-th/19607152v2.~--~1996.~--~ P.1--8.

11. Березанский Ю.М., Ус Г.Ф., Шефтель З.Г. Функциональный анализ. --К.: Выща школа. ~--~1990.~--~600~с.

12. Кужель А.В. Классификация спектра в алгебрах// Докл.АН СССР. Сер.А.~--~1983. ~--~№.10.~--~С.11--14.

13. Пирс Р. Ассоциативные алгебры. --М.: Мир.~--~1986.~--~543 с.

	Spectral properties of linear operators on Hilbert quaternionic bimodules (in Russian)
Author	I.I.Karpenko, D.L.Tyshkevich V. I. Vernadsky Taurida National University
Abstract	In the paper, besides classical systems of the axioms accepted in the theory of bimodules over associative rings and in the theory of Hilbert quaternionic modules, the additional axioms taking place in the most common realization of Hilbert modules are offered. Independence of the given axioms of existing systems and also their importance for effective work with Hilbert bimodules are shown. We offer a classification of spectrum of a linear operator over Hilbert quaternionic bimodule which is completely coordinated to classification of spectrum of a complex linear operator. At the same time this classification gives some new properties of the spectrum.
Keywords	spectral properties, linear operator, Hilbert quaternionic bimodule
DOI	doi:10.30970/ms.30.1.67-82
Reference	1. Березин А.В., Курочкин Ю.А., Толкачев Е.А. Кватернионы в релятивистской физике.--М.: Едиториал УРСС.~--~2003.~--~200 с. 2. De Leo S., Ducati G. Quaternionic differential operators// arXiv:math-ph/0005023v3.~--~2002.~--~P.1--25. 3. De Leo S., Rotelly P. Translation between Quaternion and Complex Quantum Mechanics// arXiv:hep-th/9401009 v1.~--~1994.~--~P.1--15. 4. Курош А.Г. Лекции по общей алгебре.-- М.: Наука.~--~1973.~--~399 с. 5. Фейс К. Алгебра: кольца, поля, модули и категории, т.1,2. -- М.: Мир.~--~1977. ~--~688,463 с. 6. De Leo S., Scolarici G. Right eigenvalue equation in quaternionic quantum mechanics// Journal Phisics. A33.~--~2000.~--~P.2971--2995. 7. De Leo S., Scolarici G., Solombino L. Quaternionic eigenvalue problem// J.Math. Phys. Bd.43.~--~2002.~--~V.11.~--~P.5815--5829. 8. Sharma C.S., Coulson T.J. Spectral theory for unitary operators on a quaternionic Hilbert space// J.Math.Phys.~--~1987. -- 28(9).~--~P.1941--1945. 9. Куратовский К. Теория множеств.--М.: Мир. ~--~1970.~--~460 с. 10. Abdel-Khalek Kh. Quaternion analysis// arXiv:hep-th/19607152v2.~--~1996.~--~ P.1--8. 11. Березанский Ю.М., Ус Г.Ф., Шефтель З.Г. Функциональный анализ. --К.: Выща школа. ~--~1990.~--~600~с. 12. Кужель А.В. Классификация спектра в алгебрах// Докл.АН СССР. Сер.А.~--~1983. ~--~№.10.~--~С.11--14. 13. Пирс Р. Ассоциативные алгебры. --М.: Мир.~--~1986.~--~543 с.
Pages	67-82
Volume	30
Issue	1
Year	2008
Journal	Matematychni Studii
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Table of content of issue	html