Geometrical Characterization of RN-operators between Locally Convex Vector Spaces

TL;DR

This paper characterizes RN-operators between locally convex vector spaces using geometric properties of bounded sets, extending previous results from Banach spaces to a broader class of spaces.

Contribution

It provides a new geometric characterization of RN-operators in locally convex spaces, generalizing known Banach space results.

Findings

Equivalence of RN-operator condition with dentability of bounded sets

Extension of geometric characterization from Banach to locally convex spaces

Provides a theorem linking operator properties with set dentability

Abstract

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.