Some Holomorphic Semi-groups.

W ∗ k = {x ǫX|xǫ Domain (A(a1) . . .A (ak)) for all a1, . . . , akǫA} is dense inX in theweak* topology. Let {e1, . . . , en} be a basis forA and setAi = A(ei), A ∗ i = A(ei). If α = (α1, . . . , αk), 1 ≤ αi ≤ n, is a sequence of integers we denote a product such as Aα1 . . .Aαk byAα; |α| = k is the length ofα. The operatorB ◦ = Σ|α|≤maαAα (the coefficients are complex numbers) is defined on Wm and denoting α in reversed order by α , the operator B̃ = Σ|α|≤maαA ∗ α is defined on W ∗ m. The closure B of B ◦ and the weak* closure B̃ of B̃ are well-defined operators. If for any real n-vector, ξ = (ξ1, . . . , ξn), |Σ|α|≤maαξα| ≥ ρ‖ξ‖ m (ρ > 0 is fixed and ‖ξ‖ is the ordinary Euclidean norm) then Theorem 1. The adjoint B of B is B̃.