On Close To Scalar Families For Fractional Evolution Equations: Zero-One Law

For {Rα,β(t)}t≥0 being a strongly continuous (α, β) -resolvent family on a Banach space, we show that the assumption supt>0‖1tβ-1Eα,β(λtα)Rα,β(t)-I‖=:θ<1 yields that R_α , β(t) = t^β – 1E_α , β(λt^α) for all t> 0 and λ≥ 0 , where E_α , β(s) denotes the Mittag–Leffler function with parameters α, β> 0. This implication is known as the zero–one law. In particular, we provide new insights on the structural properties of the theories of C-semigroups, strongly continuous cosine families, β-times integrated semigroups and α-resolvent families, among others. For β-times integrated semigroups, an example shows that in the sector {(α,β):0<α≤1;α≤β} the requirement θ< 1 is optimal: for θ= 1 the result is false.