Necessary and Sufficient Condition for the Quaternion Vector Measure

Authors: Mei Li, Fahui Zhai

Abstract:

In this paper, the definitions of the quaternion measure and the quaternion vector measure are introduced. The relation between the quaternion measure and the complex vector measure as well as the relation between the quaternion linear functional and the complex linear functional are discussed respectively. By using these relations, the necessary and sufficient condition to determine the quaternion vector measure is given.

Keywords: Quaternion, Quaternion measure, Quaternion vector measure, Quaternion Banach space, Quaternion linear functional.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1128087

References:

[1] J. A. Clarkson, Abstrakte Funktionen und lineare Operatoren, Trans. Amer. Math. Soc., Vol. 40, 1936, pp. 396-414.
[2] I. M. Gel’fand, Uniformly convex spaces, Mat. Sb. (N. S.) 4(46), 1938, pp. 235-286.
[3] R. G. Bartle, A general bilinear vector integral, Studia Math., Vol. 15, 1956, pp. 337-352.
[4] N. Dinculeanu and I. Kluv´anek, On vector measures, Proc. London Math. Soc.,17(3), 1967, pp. 505-512.
[5] N. Dunford and J. T. Schwartz, Linear operators. Part I, Interscience, New York and London, 1958.
[6] J. Lindenstrauss and A. Pelczy´nski, Absolutely summing operators in Lp-spaces and their applications, Studia Math., Vol. 29, 1968, pp. 275-326.
[7] J. Diestel and J. J. Uhl, Jr, Vector measures, Math Surveys Monographs, Providence, 1977.
[8] I. Kluv´anek, Applications of vector measures, Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, N.C.), 1980, pp.101-134.
[9] J. Diestel and J. J. Uhl, Jr, Progress in vector measuresł 1977C83, Indagationes Mathematicae, Vol. 8(1), 1997, pp.33-42.
[10] A. Fern´andez and Faranjo, Rybakov’s theorem for vector measures in Fr´echet spaces, Measure Theory and its Applications, Lecture Notes in Mathematics Vol. 1033, 1984, pp.144-192.
[11] G. P. Curbera and W. J. Ricker, Vector Measures, Integration and Applications, Positivity, Birkh¨auser Basel, 2007, pp.127-160.
[12] C. S. Sharma and T. J. Coulson, Spectral theory for unitary operators on a quaternionic Hilbert space, Journal of Mathematical Physics, Vol. 28(9), 1987, pp.1941-1946.
[13] S. H. Kulkarni, Representation of a real B∗-algebra on a quaternionic Hilbert space, Proceedings of the American Mathematical Society, Vol.121(2), 1994, pp.505-509.
[14] Chi-Keung Ng, On quaternionic functional analysis, Mathematical Proceedings of the Cambridge Philosophical Society, Vol.143(2), 2007, pp.391-406.
[15] S. V. Ladkovsky, Algebras of operators in Banach spaces over the quaternion skew field and the octonion algebra, Journal of Mathematical Sciences, Vol. 144(4), 2007, pp.4301-4366.
[16] F. Brackx, J. S. R. Chisholm and V Souˇcek, Clifford analysis and its applications, Kluwer Academic Publishers, 2001.
[17] S. L. Adler, Quaternionic quantum mechanics and quantum filed, Oxford university press, 1995.
[18] H. C. Lee, Eigenvalues of canonical forms of matrices with quaternion coefficients, Proc. R. I. A., Vol. 52, 1949, pp.253-260.
[19] W. J. Ricker, Operator Algebras Generated by Commuting Projections: A Vector Measure Approach, Springer Berlin Heidelberg, 1999.