A space of continuous E-valued functions is defined for a locally convex space E, and the generalized resolvent of an operator A on E is defined as an operator on . It is noted that may exist when the classical resolvent (λ − A)⁻¹ fails to exist. Conditions on are given that are necessary and sufficient to guarantee that A is the generator of a semigroup T(t). The action of is characterized by convolution against the semigroup, and the semigroup is computed as the limit of acting on an approximate identity.

Conditions on an operator B are introduced that are sufficient to guarantee that A + B is the generator of a semigroup whenever A is. A formula is given for the perturbed semigroup.

Two characterizations of semigroups that can be extended holomorphically into some sector of the complex plane are given. One is in terms of the growth of the derivative (d/dt) T(t) as t approaches 0, the other is in terms of the behavior of ⁿ, the powers of the generalized resolvent.

Throughout, the generalized resolvent plays a role analogous to the role of the classical resolvent in the work of Hille, Phillips, Yosida, Miyadera, and others.