I was reading the paper Sum of small numbers of idempotents where they have proved that every operator on a Hilbert space can be written as a complex linear combination of 16 projections. Which also means that positive operators are dense in $B(H)$.
I was wondering if this is true in $B_2(H)$ as well. Does the set $$S=\{T\in B_2(H) : \left<Tx,x\right>\geq 0 ~\forall x\in H\}$$ span whole of $B_2(H)$ in Hilbert Schmidt norm?