Dissipativity of the delay semigroup

We call (T (t))t≥0 an abstract delay semigroup. This notion is introduced in more detail in Section 2. The semigroup thus obtained is rarely a contraction semigroup. In this paper we will show that under very mild conditions, we may change the inner product on E into an equivalent inner product, such that the delay semigroup becomes a (generalized) contraction semigroup. A contraction semigroup is a semigroup (S(t))t≥0 such that ||S(t)|| ≤ 1 for all t ≥ 0. A generalized contraction semigroup is a semigroup (S(t))t≥0 such that ||S(t)|| ≤ e for some λ ∈ R and all t ≥ 0. As explained in Section 4.1 our results improve upon [11], where dissipativity of functional differential equations is established under stronger conditions, and upon [5, Section 10.3], where the results are restricted to linear delay equations of the form