I am interested in operators on a non-reflexive Banach space. Let $X$ be a Banach space and let $L(X)$ be the algebra of operators acting on $X$. We may embed $L(X)$ into $L(X^{\ast\ast})$ by $\Phi(T)=T^{\ast\ast}$; this is an algebra homomorphism. Is the image $\Phi(L(X))$ dense in $L(X^{**})$ in the sense of some of the classical operator topologies like WOT, SOT etc.?
$L(X^{∗∗})$instead of into$L(X^∗)$? $\endgroup$$X^*$has the bounded approximation property, then the embedding of $L(X)$ into$L(X^*)$is strongly dense. This is an immediate consequence of the principle of local reflexivity, which you can find in text books. $\endgroup$$L(X^{**})$that maps some point in $X$ to a point not in $X$. There are plenty of rank one operators that do that. Operators in $\Phi(L(X))$ map $X$ into $X$. $\endgroup$$L(X^{*})$, I was thinking about taking limits of uniformly bounded nets. If you really only care about SOT density, then no condition is needed. Given $T$ in$L(X^{*})$and a finite dimensional subspace $F$ of$X^{*}$, choose a basis $(f_n)$ for $f$ and elements $(x_n)$ in $X$ s.t. $(x_n,f_n)$ is biorthogonal and consider$S:=\sum_n x_n\otimes f_n$. This finite rank operator is clearly weak$^*$continuous and agrees with $T$ on $F$. $\endgroup$