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Feynman's Operational Calculus and BeyondNoncommutativity and Time-Ordering$
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Gerald W Johnson, Michel L. Lapidus, and Lance Nielsen

Print publication date: 2015

Print ISBN-13: 9780198702498

Published to Oxford Scholarship Online: September 2015

DOI: 10.1093/acprof:oso/9780198702498.001.0001

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Spectral Theory for Noncommuting Operators

Spectral Theory for Noncommuting Operators

(p.327) 10 Spectral Theory for Noncommuting Operators
Feynman's Operational Calculus and Beyond

Gerald W. Johnson

Michel L. Lapidus

Lance Nielsen

Oxford University Press

This chapter, on spectral theory for noncommuting operators, seeks to extend the operational calculus beyond the setting of the analytic functions in the disentangling algebra. When the Banach space is replaced with a Hilbert space, and the operators under consideration are bounded and self-adjoint, it is shown that the domain of the operational calculus (i.e. the domain of the disentangling map) is much richer than the disentangling algebra. The approach taken in this chapter relies in part on a stability result from Chapter 7, as well as on techniques from probability theory and the theory of distributions (in particular, the Paley–Wiener–Schwartz theorem). Also considered in this chapter is a simple class of “Trotter product formulas” that are suitable for Feynman’s operational calculus in the general Banach space setting.

Keywords:   spectral, Trotter, disentangling, stability

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