Quotients of F-spaces

Let X be a non-locally convex F-space (complete metric linear space) whose dual X' separates the points of X. Then it is known that X possesses a closed subspace N which fails to be weakly closed (see [3]), or, equivalently, such that the quotient space XIN does not have a point separating dual. However the question has also been raised by Duren, Romberg and Shields [2] of whether X possesses a proper closed weakly dense (PCWD) subspace N, or, equivalently a closed subspace N such that X/N has trivial dual. In [2], the space Hp ( 0 < p < l ) was shown to have a PCWD subspace; later in [9], Shapiro showed that £p (0 < p < 1) and certain spaces of analytic function have PCWD subspaces. Our first result in this note is that every separable non-locally convex F-space with separating dual has a PCWD subspace. It was for some time unknown whether an F-space with trivial dual could have non-zero compact endomorphisms. This problem was equivalent to the existence of a non-zero compact operator T.X—* Y, where X has trivial dual; for if such an operator exists, then we may suppose T has dense range and then the space X® Y has trivial dual and admits the compact endomorphism (x, y)—* (0,/Tx). Let us say that an F-space X admits compact operators if there is a non-zero compact operator with domain X. The most commonly arising spaces with trivial dual Lp (0 < p < 1), do not admit compact operators ([4]). However in [7] it was shown that the spaces Hp ( 0 < p < l ) possess quotients with trivial dual but admitting compact operators; equivalently there is a compact operator T with domain Hp whose kernel T" ^) is a PCWD subspace of Hp. The construction depended on certain special properties of Hp. However we show here that every separable non-locally convex locally bounded F-space with a base of weakly closed neighbourhoods of zero admits a compact operator whose kernel is a PCWD subspace, and thus has a quotient with trivial dual but admitting compact operators. This result applies to Hp and to any locally bounded space with a basis; in a sense there are many examples of compact endomorphisms in spaces with trivial dual. Let X be an F-space; we denote an F-norm (in the sense of [3]) defining the topology on X by ||-||. The following lemma is proved in [3] and [8].