Generators of regular semigroups

A regular semigroup (cf. [4, p. 38]) is a C0-semigroup T(̇) that has an extension as a holomorphic semigroup W(̇) in the right halfplane ℂ⁺, such that ||W(̇)|| is bounded in the unit rectangle Q:=(0, 1] [1, 1]. The important basic facts about a regular semigroup T(̇) are: (i) it possesses a boundary group U(̇), defined as the limit lim _s→0+ W(s+i) in the strong operator topology; (ii) U(̇) is a C₀-group, whose generator is iA, where A denotes the generator of T(); and (iii) W(s+it)=T(s)U(t) for all s+it ∈ ℂ⁺ (cf. Theorems 17.9.1 and 17.9.2 in [3]). The following converse theorem is proved here. Let A be the generator of a C0-semigroup T(̇). If iA generates a C0-group, U(̇), then T(̇) is a regular semigroup, and its holomorphic extension is given by (iii). This result is related to (but not included in) known results of Engel (cf. Theorem II.4.6 in [2]), Liu [7] and the author [6] for holomorphic extensions into arbitrary sectors, of C0-semigroups that are bounded in every proper subsector. The method of proof is also different from the method used in these references. Criteria for generators of regular semigroups follow as easy corollaries.