Functional Calculus for a Pair of Almost Commuting Selfadjoint Operators

Let (A, B) be a pair of almost commuting selfadjoint operators on Hilbert space. The aim of this paper is to construct a functional calculus defined on a class of functions two variables and satisfying natural properties (linearity and multiplicativity modulo the trace class). The problem is to make this class as big as possible. It is proved that it is impossible to construct such a functional calculus on the class of all continuously differentiable functions. On the other hand we find a class C of functions in two real variables and construct a functional calculus on this class which extends functional calculi constructed earlier. It is proved that for this functional calculus the trace formula of Helton-Howe holds.