Semigroups of operators and measures of noncompactness

It is observed that the perturbation class of an open semigroup in a Banach algebra is a closed two-sided ideal. Certain seminorms on the algebra of bounded operators are introduced; these seminorms induce norms on the quotient algebra modulo the ideal of compact operators. Using these seminorms and an assumption apparently weaker than the metric approximation property it is shown that semiFredholm operators have canonical projections (in the quotient algebra) that are not topological zero divisors. A sufficient condition is found that the converse be true. The special cases of subprojective and superprojective Banach spaces are studied. Some properties of essential spectrum are discussed.