On the Convergence of Unbounded Sequences of Semi-Groups
Rhonda Jo Hughes · Journal of the London Mathematical Society · 1977
The theory of semi-groups of unbounded linear operators developed by the author is applied in the case of a one-parameter family {Tt}t = 0 of closed, densely-defined linear operators acting in a Banach space X. It is assumed that there exists a family of projections {PN}N ε Z+ on X such that(i) for each x ε X, |PNx−x|→ → 0 as N → ∞, and (ii) PNPM = PM if M ⩽ N; moreover, (í) ∪nεz+ PN X ⊂ D), a suitable subspace of ∩t=o Domain(Tt), and (ií) for each t = 0, NεZ+, Tt PNx = PNTtx, for xεDomain(Tt). THEOREM. The infinitesimal generator A of {Tt}t=o is a closable, densely-defined operator which uniquely determines the semi-group {Tt}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, Tt = etB, where B is an unbounded scalar type operator with real spectrum; moreover, B = Ā.