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The Reproducing Kernel Property and Its Space: The Basics Read chapter Read first chapter

2015 | OriginalPaper | Chapter

61. Some Peculiarities of Quaternionic Linear Spaces

Authors : Maria Elena Luna-Elizarrarás, Michael Shapiro

Published in: Operator Theory

Publisher: Springer Basel

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Abstract

The aim of this paper is to provide a brief panorama of the subjects that have been considered in the development of quaternionic functional analysis. Since there are many similarities with classic functional analysis over real and complex numbers, it is focused on showing peculiarities which arise when one works with the basic objects in quaternionic situation: the notion of linearity, norm, inner product, linear operators and functionals, eigenvalues and eigenvectors.

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previous chapter Monogenic Signal Theory
next chapter Schur Analysis in the Quaternionic Setting: The Fueter Regular and the Slice Regular Case
Literature
1.
Adler, S.L.: Quaternionic Quantum Mechanics and Quantum Fields, p. 586. Oxford University Press, New York (1995)
2.
Agrawal, S., Kulkarni, S.H.: Dual spaces of quaternion normed linear spaces and reflexivity. J. Anal. 8, 79–90 (2000)MathSciNetMATH
3.
Agrawal, S., Kulkarni, S.H.: An analogue of the Riesz–representation theorem. Novi Sad J. Math. 30(1), 143–154 (2000)MathSciNetMATH
4.
5.
Alpay, D., Luna-Elizarrarás, M.E., Shapiro, M.: Normes des extensions quaternionique d’operateurs réels. C. R. Acad. Sci. Ser. I Math. 340, 639–643 (2005)CrossRefMATH
6.
Alpay, D., Luna-Elizarrarás, M.E., Shapiro, M.: On the norms of quaternionic extensions of real and complex linear mappings. Math. Methods Appl. Sci. 30, 1005–1036 (2007)MathSciNetCrossRefMATH
7.
Alpay, D., Colombo, F., Sabadini, I.: Pontryagin De Branges Rovnyak spaces of slice hyperholomorphic functions. J. d’Analyse Math. 121(1), 87–125 (2013)MathSciNetCrossRefMATH
8.
Alpay, D., Colombo, F., Sabadini, I.: Inner product spaces and Krein spaces in the quaternionic setting. In: Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes. A Collection of Papers Dedicated to Lev Sakhnovich. Operator Theory Advances and Applications. Linear Operators and Linear Systems. Birkhauser, Basel (2014)
9.
Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37(4), 823–843 (1936)CrossRef
10.
Chelidze, G., Vakhania, N.: On a quaternion valued Gaussian random variables. Stud. Univ. Babeş–Bolyai Math. 56(2), 295–298 (2011)MathSciNetMATH
11.
Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative Functional Calculus Theory and Applications of Slice Hyperholomorphic Functions. Progress in Mathematics, p. 222. Birkhauser, Basel (2011). ISBN 978-3-0348-0109-6
12.
13.
Emch, G.: Mécanique quantique quaternionienne et relativité restreinte. I. Helv. Phys. Acta 36, 739–769 (1963)MathSciNetMATH
14.
Finkelstein, D., Jauch, J.M., Speiser, D.: Quaternion quantum mechanics I, II, III. European Center for Nuclear Research, Geneva, Report 59–7, 59–11, 59–17 (1959)
15.
Ghiloni, R., Moretti, V., Perotti, A.: Continuous slice functional calculus in quaternionic Hilbert spaces. Rev. Math. Phys. 25(4), 1350006-1–1350007-83 (2013)MathSciNetCrossRef
16.
Glazman, I.M., Ljubic, J.I.: Finite–Dimensional Linear Analysis: A Systematic Presentation in Problem Form. MIT Press, Cambridge (1974)MATH
17.
Horwitz, L.P.: Some spectral properties of anti-self-adjoint operators on a quaternionic Hilbert space. J. Math. Phys. 34(8), 3405–3419 (1933)MathSciNetCrossRef
18.
Horwitz, L.P., Biedenharn, L.C.: Quaternion quantum mechanics: second quantization and gauge fields. Ann. Phys. 157, 432–488 (1984)MathSciNetCrossRefMATH
19.
Hungerford, T.M.: Algebra. Graduate Texts in Mathematics. Springer, New York (1974)
20.
21.
Luna-Elizarrarás, M.E., Shapiro, M.: Preservation of the norms of linear operator acting on some quaternionic function spaces. Oper. Theory Adv. Appl. 157, 205–220 (2005)CrossRef
22.
Luna-Elizarrarás, M.E., Shapiro, M.: On some relations between real, complex and quaternionic linear spaces. In: Begehr, H., Nicolosi, F. (eds.) More Progresses in Analysis, pp. 999–1008. World Scientific, Singapore (2009)CrossRef
23.
Morimoto, M.: Analytic Functionals on the Sphere. Translations of Mathematical Monographs, vol. 178. American Mathematical Society, Providence (1998)
24.
25.
26.
Rembielinski, J.: Quaternionic Hilbert space and colour confinement: II. The admissible symmetry groups. J. Phys. A Math. Gen. 13, 23–30 (1980)MathSciNetCrossRefMATH
27.
Sharma, C.S.: Complex structures on a real Hilbert space and symplectic structure on a complex Hilbert space. J. Math. Phys. 29(5), 1069–1078 (1988)MathSciNetCrossRef
28.
Sharma, C.S., Almeida, D.F.: Additive functionals and operators on a quaternionic Hilbert space. J. Math. Phys. 30(2), 369–375 (1989)MathSciNetCrossRefMATH
29.
Sharma, C.S., Coulson, T.J.: Spectral theory for unitary operators on a quaternionic Hilbert space. J. Math. Phys. 28(9), 1941–1946 (1987)MathSciNetCrossRefMATH
30.
Soukhomlinoff, G.A.: Über fortsetzung von linearen funktionalen in linearen komplexen räumen und linearen quaternionräumen. Mat. Sb.(N.S.)3, 353–358 (1938).
31.
Teichmüller, O.: Operatoren im Wachsschen Raum. J. Für Math. Bd. 174, 73–125 (1935)MATH
32.
Torgašev, A.: Numerical range and the spectrum of linear operators in Wachs spaces. Doctoral thesis, Faculty of Science, Beograd (1975)
33.
Torgašev, A.: On reflexivity of a quaternion normed space. Novi Sad J. Math. 27(2), 57–64 (1997)MathSciNetMATH
34.
Torgašev, A.: Dual space of a quaternion Hilbert space. Ser. Math. Inform. 14, 71–77 (1999)MATH
35.
Vakhania, N.N.: Random vectors with values in quaternion Hilbert spaces. Theory Probab. Appl. 43(1), 99–115 (1999)MathSciNetCrossRef
36.
Vakhania, N.N., Chelidze., G.Z.: Quaternion-valued Gaussian random variables. Theory Probab. Appl 54(2), 337–344 (2009)MathSciNet
37.
Viswanath, K.: Normal operations on quaternionic Hilbert spaces. Trans. Am. Math. Soc. 162, 337–350 (1971)MathSciNet
Metadata
Title
Some Peculiarities of Quaternionic Linear Spaces
Authors
Maria Elena Luna-Elizarrarás
Michael Shapiro
Copyright Year
2015
Publisher
Springer Basel
Book
Operator Theory

Print ISBN: 978-3-0348-0666-4

Electronic ISBN: 978-3-0348-0667-1

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-0348-0667-1_15

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