**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30761

##### Necessary and Sufficient Condition for the Quaternion Vector Measure

**Authors:**
Fahui Zhai,
Mei Li

**Abstract:**

**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.