On the Convergence of Unbounded Sequences of Semi‐Groups

The theory of semi‐groups of unbounded linear operators developed by the author is applied in the case of a one‐parameter family { T t }t = 0 of closed, densely‐defined linear operators acting in a Banach space X . It is assumed that there exists a family of projections { P N }N ε Z + on X such that(i) for each x ε X , | P N x−x |→ → 0 as N → ∞, and (ii) P N P M = P M if M ⩽ N ; moreover, (í) ∪n ε z + P N X ⊂ D ), a suitable subspace of ∩ t=o Domain( T t ), and (ií) for each t = 0, NεZ + , T t P N x = P N T t x , for x εDomain( T t ). THEOREM. The infinitesimal generator A of { T t } t =o is a closable, densely‐defined operator which uniquely determines the semi‐group {T t } t =o. A Hille‐Yosida type theorem is proved, and ā (the closure of A) is characterized as a limit of certain closed operators in X . An application to semi‐groups of unbounded scalar type operators with real spectrum is given, and it is shown that, under certain conditions, T t = e tB , where B is an unbounded scalar type operator with real spectrum; moreover, B = Ā.