mjhjmtu

Let $B=-B$ be a nowhere dense bounded closed convex set in the Hilbert space $ell_2$ such that the linear hull of $B$ is dense in $ell_2$.

Question. Is there a non-compact linear bounded operator $T:ell_2to ell_2$ such that $T(B)$ is compact?

Your revised assumption is that the norm (rather than semi-norm after the revision) on $ell_2$ given by sup-ing against vectors in $B$ is not equivalent to the usual norm on any finite codimensional subspace. So there is an ON sequence $x_n$ in $ell_2$ s.t. $sup langle x_n,brangle : bin B to 0$ as $nto infty$. Take for $T$ the orthogonal projection onto the closed span of the $x_n$.

No, take $B$ to be the set of vectors in the unit ball of $l^2$ whose first coordinate is zero. If $T: l^2 to l^2$ is a bounded operator and $T(B)$ is compact, then letting $C$ be the set of vectors in the unit ball which are nonzero only in the first coordinate (plus the zero vector), we have that $C$ is compact and therefore $T(C)$ is also compact. Thus $T(B+C) = T(B) + T(C)$ is compact, and $B + C$ contains the unit ball, so $T$ is a compact operator.