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

DOI:
https://doi.org/10.1090/S0002-9939-1994-1212283-6

MathSciNet review:
1212283

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[B]**Philip Brenner,*The Cauchy problem for symmetric hyperbolic systems in 𝐿_{𝑝}*, Math. Scand.**19**(1966), 27–37. MR**0212427**, https://doi.org/10.7146/math.scand.a-10793**[D-U]**J. Diestel and J. J. Uhl Jr.,*Vector measures*, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR**0453964****[G-J]**James Glimm and Arthur Jaffe,*Quantum physics*, Springer-Verlag, New York-Berlin, 1981. A functional integral point of view. MR**628000****[I1]**Takashi Ichinose,*Path integral for the Dirac equation in two space-time dimensions*, Proc. Japan Acad. Ser. A Math. Sci.**58**(1982), no. 7, 290–293. MR**682685****[I2]**Takashi Ichinose,*Path integral for a hyperbolic system of the first order*, Duke Math. J.**51**(1984), no. 1, 1–36. MR**744285**, https://doi.org/10.1215/S0012-7094-84-05101-9**[J1]**Brian Jefferies,*Processes associated with evolution equations*, J. Funct. Anal.**91**(1990), no. 2, 259–277. MR**1058972**, https://doi.org/10.1016/0022-1236(90)90144-A**[J2]**Brian Jefferies,*On the additivity of unbounded set functions*, Bull. Austral. Math. Soc.**45**(1992), no. 2, 223–236. MR**1155480**, https://doi.org/10.1017/S0004972700030082**[J-O]**Brian Jefferies and Susumu Okada,*Pettis integrals and singular integral operators*, Illinois J. Math.**38**(1994), no. 2, 250–272. MR**1260842****[K]**I. Kluvánek,*Operator valued measures and perturbations of semigroups*, Arch. Rational Mech. Anal.**81**(1983), no. 2, 161–180. MR**682267**, https://doi.org/10.1007/BF00250650**[R]**Gerald Rosen,*Feynman path summation for the Dirac equation: an underlying one-dimensional aspect of relativistic particle motion*, Phys. Rev. A (3)**28**(1983), no. 2, 1139–1140. MR**711511**, https://doi.org/10.1103/PhysRevA.28.1139**[S]**Laurent Schwartz,*Radon measures on arbitrary topological spaces and cylindrical measures*, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1973. Tata Institute of Fundamental Research Studies in Mathematics, No. 6. MR**0426084****[Si]**Barry Simon,*Functional integration and quantum physics*, Pure and Applied Mathematics, vol. 86, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR**544188****[Z]**T. Zastawniak,*Path integrals for the Dirac equation—some recent developments in mathematical theory*, Stochastic analysis, path integration and dynamics (Warwick, 1987) Pitman Res. Notes Math. Ser., vol. 200, Longman Sci. Tech., Harlow, 1989, pp. 243–263. MR**1020072**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
47N50,
28C20,
46N50,
81S40

Retrieve articles in all journals with MSC: 47N50, 28C20, 46N50, 81S40

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1994-1212283-6

Article copyright:
© Copyright 1994
American Mathematical Society