A New Characterization of B-bounded Semigroups with Application to Implicit Evolution Equations

Definition 1.1. Let X and Z be Banach spaces and suppose that A and B are two linear operators such that (i) A :D(A)→X, B :D(B)→ Z, with D(A), D(B)⊂X, (ii) D(A) ⊂ D(B) and for some ω ∈ R the resolvent set of A satisfies ρ(A) ⊃ ]ω,∞[. A one-parameter family of linear operators (Y (t))t≥0, which satisfies (1) Y (t) : → Z, with X ⊃ ⊇D(B), and for any t ≥ 0 and f ∈D(B) ∥∥Y (t)f ∥∥ ≤M exp(ωt)‖Bf ‖, (1.1)