ABSTRACT
The failure of the spectral mapping theorem σ ( T ( t ) ) \ { 0 } = e t σ ( A ) , t ≤ 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/eqnsmt.tif"/> for a strongly continuous semigroup (T(t)) t ≥0 with generator A on a Banach space X is the main reason for many difficulties arising in the investigation of the asymptotic behavior of these semigroups (see [5], Chapter IV and V). To overcome some of these difficulties, the essential spectrum σess (T(t)) has been used by many authors and for many concrete equations. More recently, the critical spectrum σcrit (T(t)), smaller than σess (T(t)), has been introduced in [9] and then applied in [3] to perturbed semigroups.
