Geometrical Characterization of RN-operators between Locally Convex Vector Spaces

For locally convex vector spaces (l.c.v.s.) $E$ and $F$ and for linear and continuous operator $T:E\to F$ 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 $ϵ>0$ there exists an $x\in B$ so that $ x\notin \overline{co} (B\setminus T^{-1}(T(x)+\epsilon V))$(respectively,sothat$ x\notin s$-$co (B\setminus T^{-1}(T(x)+\epsilon V)),$respectively,sothat$ x\notin {co} (B\setminus T^{-1}(T(x)+\epsilon V)) ). $Moreover,$B$iscalled$T$-dentable(respectively,$T$-s-dentable,$T$-f-dentable)ifitis$T$-V-dentable(respectively,$T$-V-s-dentable,$T$-V-f-dentable) for every absolutely convex neighborhood $V$ of zero in $F\mathrm{.}$ 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 $T:E\to F$ 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.