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Characterization of anticommutativity of self-adjoint operators in connection with Clifford algebra and applications. (English) Zbl 0810.47019

Summary: A new characterization of anticommutativity of (unbounded) self-adjoint operators is presented in connection with Clifford algebra. Some consequences of the characterization and applications are discussed.

MSC:

47B25 Linear symmetric and selfadjoint operators (unbounded)
47N50 Applications of operator theory in the physical sciences
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
15A66 Clifford algebras, spinors
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[1] A. Arai, Path integral representation of the index of Kähler-Dirac operators on an infinite dimensional manifold,J. Funct. Anal. 82 (1989), 330-369. · Zbl 0684.58010
[2] A. Arai, Supersymmetric embedding of a model of a quantum harmonic oscillator interacting with infinitely many bosons,J. Math. Phys. 30 (1989), 512-520. · Zbl 0674.58042
[3] A. Arai, A general class of infinite dimensional Dirac operators and path integral representation of their index,J. Funct. Anal. 105 (1992), 342-408. · Zbl 0803.46082
[4] A. Arai, Commutation properties of anticommuting self-adjoint operators, spin representation and Dirac operators,Integr. Equat. Oper. Th. 16 (1993), 38-63. · Zbl 0793.47020
[5] A. Arai, Momentum operators with gauge potentials, local quantization of magnetic flux, and representation of canonical commutation relations,J. Math. Phys. 33 (1992), 3374-3378. · Zbl 0787.47040
[6] A. Arai, Properties of the Dirac-Weyl operator with a strongly singular gauge potential,J. Math. Phys. 34 (1993), 915-935. · Zbl 0809.47057
[7] O. Bratteli and D.W. Robinson,Operator Algebras and Quantum Statistical Mechanics II, Springer, New York, 1979. · Zbl 0421.46048
[8] J.E. Gilbert and M.A.M. Murray,Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge University Press, Cambridge, 1991. · Zbl 0733.43001
[9] S. Pedersen, Anticommuting self-adjoint operators,J. Funct. Anal. 89 (1990), 428-443. · Zbl 0704.47020
[10] M. Reed and B. Simon,Methods of Modern Mathematical Physics Vol. I, Academic Press, New York, 1972. · Zbl 0242.46001
[11] M. Reed and B. Simon,Methods of Modern Mathematical Physics Vol. II, Academic Press, New York, 1975. · Zbl 0308.47002
[12] Yu.S. Samoilenko,Spectral Theory of Families of Self-Adjoint Operators, Kluwer Academic Publishers, Dordrecht, 1991.
[13] K. Schmüdgen,Unbounded Operator Algebras and Representation Theory, Birkhäuser, Basel, 1990.
[14] F.-H. Vasilescu, Anticommuting self-adjoint operators,Rev. Roum. Math. Pures et Appl. 28 (1983), 77-91. · Zbl 0525.47017
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