Meta-Invariant Operators over Cayley-Dickson Algebras and Spectra

S. V. Ludkovsky

Department of Applied Mathematics, Moscow State Technical University, Moscow, Russia.

DOI: 10.4236/apm.2013.31008                

Abstract

A class of meta-invariant operators over Cayley-Dickson algebra is studied. Their spectral theory is investigated. More- over, theorems about spectra of generalized unitary operators and their semigroups are demonstrated.

Keywords

Hypercomplex Numbers; Cayley-Dickson Algebras; Operator; Operator Algebra; Spectra; Spectral Measure

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. Connes, “Non-Commutative Geometry,” Academic Press, San Diego, 1994.
[2]	F. van Oystaeyen, “Algebraic Geometry for Associative Algebras,” Lecture Notes in Pure and Applied Mathemematics, Vol. 232, Marcel Dekker, New York, 2000.
[3]	G. Emch, “Mèchanique Quantique Quaternionienne et Relativitè Restreinte,” Helvetica Physica Acta, Vol. 36, 1963, pp. 739-788.
[4]	S. V. Ludkovsky, “Generalized Geometric Loop Groups of Complex Manifolds, Gaussian Quasi-Invariant Measures on Them and Their Representations,” Journal of Mathematical Sciences, Vol. 122, No. 1, 2004, pp. 2984-3010. doi:10.1023/B:JOTH.0000029575.92350.db
[5]	S. V. Ludkovsky, “Poisson Measures for Topological Groups and Their Representations,” Southeast Asian Bulletin of Mathematics, Vol. 25, No. 4, 2002, pp. 653-680. doi:10.1007/s100120200010
[6]	S. V. Ludkovsky, “Wrap Groups of Connected Fiber Bundles: Their Structure and Cohomologies,” International Journal of Mathematics, Game Theory and Algebra, Vol. 19, No. 1-2, 2009, pp. 53-128.
[7]	S. V. Ludkovsky, “Affine and Wrap Algebras over Octonions,” Journal of Mathematical Sciences, Vol. 167, No. 6, 2010, pp. 767-809. doi:10.1007/s10958-010-9960-7
[8]	S. V. Ludkovsky, “Wrap Groups of Fiber Bundles over Quaternions and Octonions,” CUBO A Mathematical Journal, Vol. 11, No. 1, 2009, pp. 101-121.
[9]	H. B. Lawson and M.-L. Michelson, “Spin Geometry,” Princeton University Press, Princeton, 1989.
[10]	S. V. Ludkovsky, “Functions of Several Cayley-Dickson Variables and Manifolds over Them,” Journal of Mathematical Sciences, Vol. 141, No. 3, 2007, pp. 1299-1330. doi:10.1007/s10958-007-0043-3
[11]	A. G. Kurosh, “Lectures on General Algebra,” Nauka, Moscow, 1973.
[12]	S. V. Ludkovsky, “Algebras of Operators in Banach Spaces over the Quaternion Skew Field and the Octonion Algebra,” Journal of Mathematical Sciences, Vol. 144, No. 4, 2008, pp. 4301-4366. doi:10.1007/s10958-007-0273-4
[13]	S. V. Ludkovsky, “Analysis over Cayley-Dickson Numbers and Its Applications,” LAP Lambert Academic Publishing, Saarbrücken, 2010.
[14]	R. Engelking, “General Topology,” Heldermann, Berlin, 1989.
[15]	L. Narici and E. Beckenstein, “Topological Vector Spaces,” Marcel-Dekker, New York, 1985.
[16]	S. V. Ludkovsky, “Differentiable Functions of Cayley-Dickson Numbers and Line Integration,” Journal of Mathematical Sciences, Vol. 141, No. 3, 2007, pp. 1231-1298. doi:10.1007/s10958-007-0042-4
[17]	A. Kolmogorov and S. Fomine, “éléments de la Théorie des Fonctions et de L’analyse Fonctionnelle,” Ellipses, Paris, 1994.
[18]	S. V. Ludkovsky, “Algebras of Operators in Banach Spaces over the Quaternion Skew Field and the Octonion Algebra,” Journal of Mathematical Sciences, Vol. 144, No. 4, 2008, pp. 4301-4366. doi:10.1007/s10958-007-0273-4
[19]	N. Dunford and J. C. Schwartz, “Linear Operators,” Wiley and Sons, Inc., New York, 1966.
[20]	L. E. Dickson, “The Collected Mathematical Papers,” Vol. 1-5, Chelsea Publishing Co., New York, 1975.
[21]	K. Gürlebeck and W. Sprossig, “Quaternionic and Clifford Calculus for Physicists and Engineers,” John Wiley and Sons, Chichester, 1997.
[22]	K. Gürlebeck and W. Sprossig, “Quaternionic Analysis and Elliptic Boundary Value Problem,” Birkhauser, Basel, 1990. doi:10.1007/978-3-0348-7295-9
[23]	R. V. Kadison and J. R. Ringrose, “Fundamentals of the Theory of Operator Algebras,” Academic Press, New York, 1983.
[24]	I. L. Kantor and A. S. Solodovnikov, “Hypercomplex Numbers,” Springer-Verlag, Berlin, 1989. doi:10.1007/978-1-4612-3650-4
[25]	S. V. Ludkovsky and F. van Oystaeyen, “Differentiable Functions of Quaternion Variables,” Bulletin des Sciences Mathematiques (Paris) Series 2, Vol. 127, 2003, pp. 755-796.
[26]	S. V. Ludkovsky, “Stochastic Processes on Geometric Loop Groups, Diffeomorphism Groups of Connected Manifolds, Associated Unitary Representations,” Journal of Mathematical Sciences, Vol. 141, No. 3, 2007, pp. 1331-1384. doi:10.1007/s10958-007-0044-2
[27]	S. V. Ludkovsky, “Differential Equations over Octonions,” Advances in Applied Clifford Algebras, Vol. 21, No. 4, 2011, pp. 773-797. doi:10.1007/s00006-011-0286-4
[28]	S. V. Ludkovsky and W. Sproessig, “Ordered Representations of Normal and Super-Differential Operators in Quaternion and Octonion Hilbert Spaces,” Advances in Applied Clifford Algebras, Vol. 20, No. 2, 2010, pp. 321-342. doi:10.1007/s00006-009-0180-5
[29]	S. V. Ludkovsky and W. Spr?ssig, “Spectral Theory of Super-Differential Operators of Quaternion and Octonion Variables,” Advances in Applied Clifford Algebras, Vol. 21, No. 1, 2011, pp. 165-191. doi:10.1007/s00006-010-0238-4
[30]	S. V. Ludkovsky and W. Sprossig, “Spectral Representations of Operators in Hilbert Spaces over Quaternions and Octonions,” Complex Variables and Elliptic Equations, Vol. 57, No. 2, 2012, pp. 1301-1324. doi:10.1080/17476933.2010.538845