Semigroups affiliated with algebras of operators

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