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An operator bound related to Feynman-Kac formulae

Author: Brian Jefferies
Journal: Proc. Amer. Math. Soc. 122 (1994), 1191-1202
MSC: Primary 47N50; Secondary 28C20, 46N50, 81S40
MathSciNet review: 1212283
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Abstract: Those Fourier matrix multiplier operators which are convolutions with respect to a matrix valued measure are characterised in terms of an operator bound. As an application, the finite-dimensional distributions of the process associated with Dirac equation are shown to be unbounded on the algebra of cylinder sets.

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  • [B] P. Brenner, The Cauchy problem for symmetric hyperbolic systems in $ {L_p}$, Math. Scand. 19 (1966), 27-37. MR 0212427 (35:3299)
  • [D-U] J. Diestel and J. J. Uhl, Jr., Vector measures, Math. Surveys Monographs, vol. 15, Amer. Math. Soc., Providence, RI, 1977. MR 0453964 (56:12216)
  • [G-J] J. Glimm and A. Jaffe, Quantum physics: A functional integral point of view, Springer-Verlag, New York, 1981. MR 628000 (83c:81001)
  • [I1] T. Ichinose, Path integral for the Dirac equation in two space-time dimensions, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), 290-293. MR 682685 (84i:81036)
  • [I2] -, Path integral for a hyperbolic system of first order, Duke Math. J. 51 (1984), 1-36. MR 744285 (86j:35104)
  • [J1] B. Jefferies, Processes associated with evolution equations, J. Funct. Anal. 91 (1990), 259-277. MR 1058972 (93b:47140)
  • [J2] -, On the additivity of unbounded set functions, Bull. Austral. Math. Soc. 45 (1992), 223-236. MR 1155480 (93b:28037)
  • [J-O] B. Jefferies and S. Okada, Pettis integrals and singular integral operators, Illinois J. Math. 38 (1994). MR 1260842 (94m:47103)
  • [K] I. Kluvánek, Operator valued measures and perturbations of semi-groups, Arch. Rational Mech. Anal. 81 (1983), 161-180. MR 682267 (84j:28019)
  • [R] G. Rosen, Feynman path summation for the Dirac equation: An underlying one-dimensional aspect of relativistic particle motion, Phys. Rev. A (3) 28 (1983), 1139-1140. MR 711511 (84i:81038)
  • [S] L. Schwartz, Radon measures in arbitrary topological spaces, Oxford Univ. Press., Tata Institute of Fundamental Research, Bombay, 1973. MR 0426084 (54:14030)
  • [Si] B. Simon, Functional integration and quantum physics, Academic Press, New York, San Francisco, and London, 1979. MR 544188 (84m:81066)
  • [Z] T. Zastawniak, Path integrals for the Dirac equation--some recent developments in the mathematical theory, Stochastic Analysis, Path Integration and Dynamics (K. D. Elworthy and J-C Zambrini, eds.), Pitman Res. Notes in Math., vol. 200, Longman Sci. Tech., Harlow, 1989, pp. 243-263. MR 1020072 (90j:81057)

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