I am interested in 'algebraic-density'-type properties of second adjoint operators in the algebra of bounded operator on a second dual of a Banach space. Incidentally, I have a problem with justification of the following question:

Say we have a non-quasi reflexive Banach space $V$ (that is $V^{\prime\prime}/V $ is infinite-dimensional).

Pick $y^{\prime\prime}\in V^{\prime\prime} \setminus V$. Is there a (left-) invertible operator $S\colon V\to V$ with $S^{\prime\prime} y^{\prime\prime} \in V$?