Distances between elements of a semigroup and estimates for derivatives

Abstract

This paper is concerned first with the behaviour of differences T (t) − T (s) near the origin, where (T (t)) is a semigroup of operators on a Banach space, defined either on the positive real line or a sector in the right half-plane (in which case it is assumed analytic). For the non-quasinilpotent case extensions of results in the published literature are provided, with best possible constants; in the case of quasinilpotent semigroups on the half-plane, it is shown that, in general, differences such as T (t) − T (2t) have norm approaching 2 near the origin. The techniques given enable one to derive estimates of other functions of the generator of the semigroup; in particular, conditions are given on the derivatives near the origin to guarantee that the semigroup generates a unital algebra and has bounded generator.

Boundary values of analytic semigroups and associated norm estimates

“…Thus, using the classical Ahlfors-Heins theorem [5, pp. 115-116], we apply these results to show that if (T (t)) t∈C + is a quasinilpotent semigroup on the open half plane, and if there exists δ > 0, µ > 0 and λ > 0 such that sup n∈Z λ |n| T (δ + inµ) < +∞, then we have, for every y ≥ 0 and every h > 0, Some very general results for the non-quasinilpotent case may be found in [3].…”

Lower estimates near the origin for functional calculus on operator semigroups

“…More importantly, the theorem applies to many other examples, such as dμ(t) = (χ [1,2] − χ [2,3] )(t)dt and μ = δ 1 − 3δ 2 + δ 3 + δ 4 , which are not accessible with the methods of [1,3,4,6]. …”

“…For example, if T (t) −T (2t) < 1/4 on an interval (0, t 0 ), then, roughly speaking, (T (t)) t>0 has a bounded infinitesimal generator (see [1]). …”

Dichotomy Results for Norm Estimates in Operator Semigroups

“…these include µ = δ 1 −δ 2 , the difference of two Dirac measures, where F (s) := Lµ(s) = e −s − e −2s and F (−sA) = T (t) − T (2t). More importantly, the theorem applies to many other examples, such as dµ(t) = (χ [1,2] − χ [2,3] )(t)dt and µ = δ 1 − 3δ 2 + δ 3 + δ 4 , which are not accessible with the methods of [1,9,10,14].…”