A non-commutative joint spectral theory

Abstract

For certain m-tuples $a = {a_1, ... ,a_n)$ of elements $a_j$ in a unital Ba11ach algebra, we construct a joint spectrum $\\gamma(a)$ and a functional calculus with a spectral mapping theorem. It is not assumed that the $a_j$ j commute but rather that they commute modulo the Jacobson radical of the algebra they generate. For matrices, this last condition is equivalent to their being simultaneously triangularizable. This work extends that of M.E. Taylor, R.F.V. Anderson, and A. Mcintosh and A. Pryde.