  In this paper we present a new extension of the theory of well-bounded
operators to cover operators with complex spectrum. In previous work a new
concept of the class of absolutely continuous functions on a nonempty compact
subset $\sigma$ of the plane, denoted $AC(\sigma)$, was introduced. An
$\AC(\sigma)$ operator is one which admits a functional calculus for this
algebra of functions. The class of $AC(\sigma)$ operators includes all of the
well-bounded operators and trigonometrically well-bounded operators, as well as
all scalar-type spectral operators, but is strictly smaller than Berkson and
Gillespie's class of $AC$ operators. This paper develops the spectral
properties of $AC(\sigma)$ operators and surveys some of the problems which
remain in extending results from the theory of well-bounded operators.
