Operators factoring through a generalized sequence space; applications
By NICOLE DE GRANDE-DE KIMPE of Brussel (Eingegangen am 10.7. 1978)
Q 1. Introduction
The results obtained in this paper belong to the theory of BANACH operator ideals as well as to the theory of locally convex spaces with a SCHAUDER basis. However both types of results are based on the same factorization techniques. In this case we consider operators which factor through a generalized sequence space. The general basic properties of operators into a generalized sequence space are given i n Q 2.
In 5 3 we apply the results of Q 2 in order to obtain, with unified proofs, a large class of BANACH operator ideals. Some of them have been studied separately in the literature. In Q 4 the results of Q 2 are used in order to obtain information on the structure of a locally convex space X having a A-basis (xi, fi) by factoring the identity operator on X through the generalized sequence space A ( X ) . Doing so we obtain, again with unified proofs new (and old) results on spaces with an absolute basis ( A = P). The case of a A-basis ( A =+= 1 1 ) is especially interesting in the case where A is nuclear. This case contains e.g. the case where the basis "decreases rapidly".
-If not mentioned otherwise X , Y , Z will denote sequentially complete, locally convex, HAUSDORFP spaces. A fundamental system of barrelled zeroneighbourhoods of the origin of X will be denoted by 'Vx and the corresponding semi-norms on X by p , (UE"IRx). By ax we mean a fundamental system of closed, absolutely convex, bounded subsets of X. For BESD, we denote by X B the associated BANACH space, normed by IIXII,=inf {A I xEAB}. The symbols X,(Xi) stand for X with the weak topology a(X, X') (a(X', X)). The space X' equipped with the strong topology /?(X', X ) is written as X i and its seminorms by p B If every B โฌ a X is relatively compact in X, then X is called a MONTEL space.
If X has the MACKEY topology t ( X , X') and X i is sequentially complete then X is called a G-space (see [4]).
-By A we always mean a perfect sequence space with a-dual space Ax. For elementary facts on sequence spaces, we refer to [8]. Recall that the normal topology on A is determined by the seminorms p(yi, ( (&)