The sets of monomorphisms and of almost open operators between locally convex spaces (Q2750866)

The authors characterize when the set of monomorphisms and the set of almost open operators are open in the set of all operators \(L(E,F)\) between two locally convex spaces \(E\) and \(F\) for the topology of uniform convergence on the bounded subsets of \(E\). NEWLINENEWLINENEWLINEBounded below and almost open continuous linear operators between normed spaces, their relation with the topological divisors of zero in the normed algebra of all operators, and the approximate point spectrum have been extensively studied [see \textit{S. K. Berberian}, ``Lectures in functional analysis and operator theory'', Springer, New York (1974; Zbl 0296.46002) (Sections 56, 57), \textit{R. E. Harte}, ``Invertibility and singularity for bounded linear operators'', New York (1988; Zbl 0636.47001), and the articles of \textit{Y. A. Abramovich, C. D. Aliprantis} and \textit{I. A. Polyrakis}, Atti Sem. Mat. Fis. Univ. Modena 44, No. 2, 455-464 (1996; Zbl 0867.47001) and \textit{R. E. Harte}, Proc. Am. Math. Soc. 90, No. 2, 243-249 (1984; Zbl 0541.46005)]. In [\textit{Y. A. Abramovich, C. D. Aliprantis} and \textit{I. A. Polyrakis}, op. cit. (Proposition 2.2)] it is shown that the set of bounded below operators (or monomorphisms) between two normed spaces \(X\) and \(Y\) is open in the normed space of operators \(L(X,Y)\). The corresponding result for almost open operators can be seen in \textit{R. E. Harte}'s book [op. cit. (Theorem 3.4.3)].NEWLINENEWLINENEWLINEThese results are extensions of the well-known fact that the set of isomorphisms from a Banach space \(X\) onto a Banach space \(Y\) is an open subset of the Banachspace \(L(X,Y)\), a result which can be proved using the Neumann series for a linear operator. The recent article by \textit{P. G. Casazza} and \textit{N. J. Kalton} [Proc. Am. Math. Soc. 127, No. 2, 519-527 (1999; Zbl 0916.47013)] deals with extensions of Paley-Wiener perturbation theory. In the case of continuous linear operators from a locally convex space \(E\) into itself, it was proved by Kasahara in 1972 that if the set of isomorphisms is open in the space of operators \(L_b(E)\) endowed with the topology of uniform convergence on the bounded subsets of \(E\), then the space \(E\) must be normable. If \(E\) is complete, in the terminology of topological algebras, \(L_b(E)\) is a \(Q\)-algebra if and only if \(E\) is normable. NEWLINENEWLINENEWLINEIn the interesting paper under review, the authors treat for the first time the case when the domain and range spaces are different, within the context of general locally convex spaces. They show that, when the set of monomorphisms is not empty, it is open if and only if the domain space is normable. When the domain is quasinormable and the range is a normed space, they also obtain that the set of almost open operators is open. Related results for the sets of surjective and open operators are also obtained.