Feynman's Operational Calculi for Noncommuting Operators: Spectral Theory

Abstract

In recent papers the authors presented the key ideas involved in their approach to Feynman's operational calculi for a system of not necessarily commuting bounded linear operators acting on a Banach space. The central objects of the theory are the disentangling algebra, a commutative Banach algebra, and the disentangling map which carries this commutative structure into the noncommutative algebra of operators. Under assumptions concerning the growth of disentangled exponential expressions, the associated functional calculus for the system of operators is a distribution with compact support which we view as the joint spectrum of the operators with respect to the disentangling map. The spectral properties of the disentangling maps are studied in this paper.

Stability Properties for Feynman’s Operational Calculus in the Combined Continuous/Discrete Setting

“…, r n ) is a commutative Banach algebra with norm given by Equation (5). (One can find a proof of this statement in the paper [6]. )…”

“…We will often refer to the Banach algebras defined above as A and D. It is shown in [6] that these algebras can, in fact, be identified. Now associate to each of the operators A j , j = 1, .…”

“…. , n where µ j is continuous and η j is purely discrete and is written in the form of Equation (8). The action of the disentangling map T λ 1 ,...,λ n :…”

“…he was aware of how difficult it would be to make his operator calculus mathematically rigorous. One approach used to make Feynman's operational calculus rigorous was initiated by Jefferies and Johnson in [6][7][8][9] where the time-independent calculus was introduced and in [10] where the time-dependent operational calculus was introduced. Further work on this approach to Feynman's calculus has been done by Johnson and Kim in [11], the present author in [21], and Johnson and Nielsen in [14] and [15].…”

“…(See the paper [2] for the disentangling of exponential functions. See also the paper [10] for connections between the operational calculus as developed in [6][7][8][9] and the operational calculus for the exponential function seen in [2].) One sees, in [20], that the use of exponential functions makes stability theorems somewhat easier to obtain, especially in the time-dependent setting where no measurability problems arise.…”