Semigroups affiliated with algebras of operators

If $ \mathcal{B} $ is a Banach algebra of bounded linear operators on a Banach space X, and if A is a closed operator on X such that $ (\lambda - A)^{-1}$ is in $\mathcal{B}$ for some $ \lambda \in \mathbb{C}$, then A is said to be affiliated with $ \mathcal{B} $. This paper examines when such an operator generates a $ C_0 $ or an integrated semigroup $ \{T(t)\}_{t \geq 0} $ in $ \mathcal{B} $. The spectral and essential spectral properties of A and $ \{T(t)\}_{t \geq 0} $ relative to $ \mathcal{B} $ are also studied. A number of consequences involving specific algebras are included.