Feynman's operational calculi for noncommuting operators: Definitions and elementary properties

An approach to Feynman's operational calculus is given for systems of bound ed, not necessarily commuting, linear operators acting on a Banach space. A commuting Banach algebra, the 'disentangling algebra', is constructed in w hich functions of formal objects associated with the operators can be forme d via the usual commutative functional calculus. The disentangling of the c ommutative representation of tile system is implemented by a system of meas ures which keep track of the time ordering. A change in the system of measu res may induce a different functional calculus-hence the wc,rd "'calculi" i n the title of tile pai,cr. We introduce the key ideas in this paper and es tablish some of the basic properties of these calculi.