Transferring locally uniformly convex norm by bounded linear operator from one Banach space to another

Question

I'm trying to find a simple proof this theorem

Let $Y$ be a locally uniformly convex Banach space and $X$ be a Banach space and let $T$ be a bunded linear operator from $X$ into $Y$ such that for every bounded sequence $(x_n)_{n=1}^{\infty}$ in $X$ with $\|Tx_n-Tx\|\to 0$ we have that $$x\in \overline {span (x_n)_{n=1}^{\infty}}^{\|.\|}$$ (in particular, whenever $x\in \overline {conv (x_n)_{n=1}^{\infty}}^{\|.\|}$ , or weak-$\lim x_n=x$). Then $X$ is locally uniformly convex renormable.

I think we need to use an equivalent norm as follow $$|x|=\|x\|+\|Tx\| \ \ \ \text{for all }x\in X$$ Is it correct? Any help will be appreciated! Thanks.

Answer 1

Transfer techniques is presented in Section 2 of Chapter II and in Section 2 of Chapter VII of the book Deville-Godefroy-Zizler, Smoothness and renormings in Banach spaces. Possibly some of this techniques can be adjusted to cases you are interested in.