# Second adjoint operators on non quasi-reflexive Banach spaces

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 $S^{\prime\prime} y^{\prime\prime} \in V$?

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This is very similar to a math.stackexchange question: math.stackexchange.com/questions/99281/second-adjoint-operators/… My answer their is easily adapted to show that the answer to this question is "no". – Matthew Daws Jan 29 '12 at 18:05

You want $S$ to be an isomorphism from $V$ onto a subspace, say $X$. (You also want $X$ to be complemented, but that is irrelevant for the answer.) This implies that $S^{''}$ is an isomorphism from $V^{''}$ onto
$X^{\perp\perp} \subset V^{''}$. So no operator like you want exists on any non reflexive Banach space.
Nitpick: you need $V''$ not $V$ in the 2nd line. – Matthew Daws Jan 29 '12 at 20:34