Geometrical Characterization of RN-operators between Locally Convex Vector Spaces

• Published in: arXiv.org 2015-02
• Main Authors: Reinov, Oleg, Asfand Fahad
• Format: Article
• Language: English
• Subjects: Banach spaces; Operators (mathematics); Theorems; Vector spaces
• ISSN: 2331-8422

For locally convex vector spaces (l.c.v.s.) \(E\) and \(F\) and for linear and continuous operator \(T: E \rightarrow F\) and for an absolutely convex neighborhood \(V\) of zero in \(F\), a bounded subset \(B\) of \(E\) is said to be \(T\)-V-dentable (respectively, \(T\)-V-s-dentable, respectively, \(T\)-V-f-dentable) if for any \(\epsilon>0\) there exists an \(x\in B\) so that \( x\notin \overline{co} (B\setminus T^{-1}(T(x)+\epsilon V))\) (respectively, so that \( x\notin s\)-\(co (B\setminus T^{-1}(T(x)+\epsilon V)),\) respectively, so that \( x\notin {co} (B\setminus T^{-1}(T(x)+\epsilon V)) ). \) Moreover, \(B\) is called \(T\)-dentable (respectively, \(T\)-s-dentable, \(T\)-f-dentable) if it is \(T\)-V-dentable (respectively, \(T\)-V-s-dentable, \(T\)-V-f-dentable) for every absolutely convex neighborhood \(V\) of zero in \(F.\) RN-operators between locally convex vector spaces have been introduced in [5]. We present a theorem which says that, for a large class of l.c.v.s. \(E, F,\) if \(T: E \rightarrow F\) is a linear continuous map, then the following are equivalent: 1) \(T \in RN(E,F);\) 2) Each bounded set in \(E\) is \(T\)-dentable; 3) Each bounded set in \(E\) is \(T\)-s-dentable; 4) Each bounded set in \(E\) is \(T\)-\(f\)-dentable. Therefore, we have a generalization of Theorem 1 in [8], which gave a geometric characterization of RN-operators between Banach spaces.