SEMIGROUPS AFFILIATED WITH ALGEBRAS OF OPERATORS

If 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 B for some lambda is-an-element-of C, then A is said to be affiliated with B. This paper examines when such an operator generates a C0 or an integrated semigroup {T(t)}t greater-than-or-equal-to 0 in B. The spe ctral and essential spectral properties of A and {T(t)}t greater-than- or-equal-to -=0 relative to B are also studied. A number of consequenc es involving specific algebras are included.