Eine Verallgemeinerung des Satzes vom abgeschlossenen Wertebereich in lokalkonvexen Räumen

Abstract

This paper extends Kato's proof [5] of Banach's closed range theorem to locally convex spaces. Thus we consider a locally convex space (E,α) and pairs (M,N) of closed subspaces. We call such a pair α-open, if and only if there exists a directed, total system Γ of seminorms generating the topology induced by a on M+N, such that the minimal gap γ_p(M,N)>O for each pεΓ. Our main result is a generalisation of the closed range theorem and it consists of statements on relationships between the following properties: (a) M+N α-closed, (b) M^⊥+N^⊥ β (E′,E)-closed, (c) M^⊥+N^⊥ σ (E′,E)-closed, (d) (M,N) α-open, (e) (M,N) σ (E,E′)-open, (f) (M^⊥,N^⊥) β(E′,E)-open, (g) (M^⊥,N^⊥) σ(E′,E)-open, (h) M+N=(M^⊥∩N^⊥)^⊥, (i) M^⊥+N^⊥=(M∩N)^⊥.

By specialising the space (E,α) and the subspaces M,N, our generalisation includes the closed range theorems of Dieudonné and Schwartz [4], Browder [1] and Mochizuki [12]. It is shown that these theorems not only hold for closed linear operators but even for closed linear relations. We are therefore able to obtain “closed domain theorems” which extend Brown's examinations in Banach-spaces [2] to locally convex spaces.