The joint spectrum of commuting nonnormal operators

Abstract

We define the joint approximate spectrum of a finite family of bounded commuting operators on a Hilbert space and show that it is closed and nonempty. If the operators are all hyponormal, then the family need not be finite, and the joint approximate spectrum can be defined in terms of the multiplicative linear functionals on the generated C ∗ {C^\ast } -algebra. These definitions extend the usual definition of the joint spectrum of a family of commuting normal operators.