Effects of absolute continuity in Feynman's operational calculus

Author:
Lance Nielsen

Journal:
Proc. Amer. Math. Soc. **131** (2003), 781-791

MSC (2000):
Primary 47A60, 46H30; Secondary 47A56, 47B38

Published electronically:
July 17, 2002

MathSciNet review:
1937417

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the effects of having an absolute continuity relation between the time-ordering measures in Feynman's operational calculus. In particular, we obtain some theorems concerning the formation of functions of several noncommuting operators or operator-valued functions under specific absolute continuity assumptions on the time-ordering measures.

**1.**Brian DeFacio, Gerald W. Johnson, and Michel L. Lapidus,*Feynman’s operational calculus and evolution equations*, Acta Appl. Math.**47**(1997), no. 2, 155–211. MR**1449438**, 10.1023/A:1005773828749**2.**Richard P. Feynman,*An operator calculus having applications in quantum electrodynamics*, Physical Rev. (2)**84**(1951), 108–128. MR**0044379****3.**Jefferies, B., and Johnson, G.W., Feynman's operational calculi for noncommuting operators: definitions and elementary properties,*Russ. J. Math. Phys*.**8**(2001), to appear.**4.**-, Feynman's operational calcui for noncommuting operators: tensors, ordered support and disentangling an exponential factor,*Math. Notes***70**(2001), accepted for publication.**5.**Brian Jefferies, G. W. Johnson, and Lance Nielsen,*Feynman’s operational calculi for time dependent noncommuting operators*, J. Korean Math. Soc.**38**(2001), no. 2, 193–226. MR**1817617****6.**Gerald W. Johnson and Michel L. Lapidus,*Generalized Dyson series, generalized Feynman diagrams, the Feynman integral and Feynman’s operational calculus*, Mem. Amer. Math. Soc.**62**(1986), no. 351, vi+78. MR**849943**, 10.1090/memo/0351**7.**Gerald W. Johnson and Michel L. Lapidus,*The Feynman integral and Feynman’s operational calculus*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. Oxford Science Publications. MR**1771173****8.**G. W. Johnson and Lance Nielsen,*A stability theorem for Feynman’s operational calculus*, Stochastic processes, physics and geometry: new interplays, II (Leipzig, 1999) CMS Conf. Proc., vol. 29, Amer. Math. Soc., Providence, RI, 2000, pp. 351–365. MR**1803428****9.**Nielsen, L.,*Stability Properties for Feynman's Operational Calculus*, Ph.D. Dissertation, Mathematics, University of Nebraska, Lincoln, 1999.**10.**Nielsen, L., Stability properties of Feynman's operational calculus for exponential functions of noncommuting operators, submitted for publication.

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Additional Information

**Lance Nielsen**

Affiliation:
Department of Mathematics and Computer Science, Creighton University, Omaha, Nebraska 68178-2090

Email:
lnielsen@creighton.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-02-06592-9

Keywords:
Functional calculus,
disentangling,
operational calculus

Received by editor(s):
July 10, 2001

Received by editor(s) in revised form:
October 10, 2001

Published electronically:
July 17, 2002

Communicated by:
N. Tomczak-Jaegermann

Article copyright:
© Copyright 2002
American Mathematical Society