The Set of Bounded Below Operators and Almost Open Operators Between Locally Convex Spaces.
by
J. A. Conejero
Universidad Politécnica de Valencia
Coauthors: J. Bonet (Universidad Politécnica de Valencia)

Let E and F be two locally convex spaces whose topologies are defined by the families of continuous seminorms cs(E) and cs(F), respectively. An operator T:E --> F is called bounded below if for every p in cs(E), there is q in cs(F) such that p(x) <= q(T(x)) for all x in E, i. e. T is an isomorphism from E into T(E). An operator S:E --> F is almost open if for every neighborhood U in E there is a neighborhood V in F such that V is contained in the closure of T(U). Bounded below operators between normed spaces have been studied by Abramovich, Aliprantis and Polyrakis in [1] and by Harte in [2]. L_b(E, F) denotes the space of all continuous and linear operators endowed with the topology of the uniform convergence on the bounded subsets of E.

We suppose that these two sets of operators are not empty. The set of bounded below operators between E and F is open in L_b(E, F) if, and only if, E is normable; and the set of almost open operators between E and F is open in L_b(E, F) if, and only if, the space F is normable and every almost open operator from E to F lifts bounded sets. We see that the last condition is essential, and we study the relevance of the quasinormability of the domain space E in this context.

References.

[1] Y. A. Abramovich, C. D. Aliprantis and I. A. Polyrakis, Some remarks on surjective and bounded below operators, Atti Sem. Mat. Fis. Univ. Modena XLIV (1996) 455-464.

[2] R. Harte, Invertibility and singularity for bounded linear operators, Marcel Dekker, New York and Basel (1988).

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Date received: April 11, 2000