quasi-weakly compact operators, co-ideals of operator ideals, and Banach spaces $X$ with $X^{**}/X$ separable

Throughout, $X$ and $Y$ will denote Banach spaces with $T\in\mathcal{L}(X,Y)$ (the space of continuous linear operators between $X$ and $Y$). We define the operator $\overline{T}\in\mathcal{L}(X^{**}/X,Y^{**}/Y)$ by the rule \begin{equation}\overline{T}(x^{**}+X)=T^{**}x^{**}+Y.\end{equation} Suppose $\mathcal{J}$ is an operator ideal (see here for the definition). We define the co-ideal of $\mathcal{J}$ via components \begin{equation}\mathcal{J}^{co}(X,Y):=\left\{T\in\mathcal{L}(X,Y):\overline{T}\in\mathcal{J}(X^{**}/X,Y^{**}/Y)\right\}\end{equation} It is clear that $\mathcal{J}^{co}$ is an operator ideal. Furthermore, it's easy to check that if $\mathcal{J}(X,Y)$ is closed (as a subspace of $\mathcal{L}(X,Y)$ endowed with the operator norm) then so is $\mathcal{J}^{co}(X,Y)$. (I do not know whether the converse is true in general.)

Denote by $\mathcal{F}$, $\mathcal{K}$, and $\mathcal{X}$, the respective ideals of finite-rank operators, compact operators, and operators with separable range. We define the ideal of quasi-weakly compact operators as the co-ideal $\mathcal{F}^{co}$. This may seem like an odd name, but it makes sense given that an operator is quasi-weakly compact if and only if it factors through a quasireflexive space. (Similarly, an operator is weakly compact if and only if it factors through a reflexive space.)

The following is probably already known, but if not then it is easy to prove by using weighted $\ell_2$-direct sums of identity operators acting on quasireflexive spaces of order $n=1,2,3,\cdots$.

Proposition 1. The ideal of quasi-weakly compact operators is not norm-closed. In particular, there exists a component space $\mathcal{F}^{co}(X)$ which is not norm-closed.

As the ideal of quasi-weakly compact operators is not closed, then what exactly is its closure? I suspect the following.

Conjecture 2. The closure of the quasi-weakly compact operators is just the co-ideal of the approximable operators. In symbols: $\overline{\mathcal{F}^{co}}=(\overline{\mathcal{F}})^{co}$.

Proof idea. It is clear that $\overline{\mathcal{F}^{co}}\subseteq(\overline{\mathcal{F}})^{co}$, so it remains only to show that $(\overline{\mathcal{F}})^{co}\subseteq\overline{\mathcal{F}^{co}}$. Suppose $T\in(\overline{\mathcal{F}})^{co}(X,Y)$. Then we can find finite-rank operators $\widehat{F}_n\in\mathcal{L}(X^{**}/X,Y^{**}/Y)$ with $\widehat{F}_n\to\overline{T}$. To complete the proof, it is sufficient to find $F_n\in\mathcal{L}(X,Y)$ such that $\overline{F}_n=\widehat{F}_n$ and $\|F_n-T\|\leq\|\widehat{F}_n-\overline{T}\|$. However, I do not immediately see how to construct $F_n$.

Regardless of whether or not conjecture 2 is true, we might be more interested in co-compact operators, i.e. the ideal $\mathcal{K}^{co}$, than we would in co-approximable operators. (I suppose there is a chance that these ideals coincide, i.e. that $\mathcal{K}^{co}=(\overline{\mathcal{F}})^{co}$, although I doubt it very much.)

Question 3. Is there any literature investigating the properties of $\mathcal{K}^{co}$?

As operators factoring through quasi-reflexive spaces fail to form a closed operator ideal, we could instead consider operators factoring through a Banach space $E$ such that $E^{**}/E$ is separable. By Example 2.5 and Theorem 2.6 in [Go93], this ideal coincides with the co-ideal $\mathcal{X}^{co}$, and hence is norm-closed.

Note that $E^{**}/E$ being separable is a very natural generalization of quasireflexivity, and as such has probably already been investigated. This property probably has a special name, too. However, I can find no information on this.

Question 4. Is there any literature discussing the property of $E^{**}/E$ being separable, and/or discussing the ideal $\mathcal{X}^{co}$ of operators factoring through such spaces?

Thanks!

• From your question it is not clear whether you are aware of the constructions of James (Pacific J. Math., 1960), Lindenstrauss (Israel J. Math, 1971), Bellenot (J. Funct. Anal., 1982), and results of Valdivia (Israel J. Math. 1988). Some years ago I studied the construction of Bellenot, in particular I used it to construct an infinite inverse sequence of separable spaces with strictly singular quotient maps (Bull. Polish Acad. Sci. Math. 44 (1996), no. 2, 143–146). – Mikhail Ostrovskii May 10 '16 at 16:54
• Thank you! Indeed, I have seen none of those papers (nor even heard of them). I looked at the James paper just now, and it has the following interesting theorem: If $E$ is a space with boundedly complete basis with basis constant = 1, then there exists a Banach space $X$ with shrinking basis such that $X^{**}=X+E_1$ for some isometric copy $E_1$ of $E$. I think I see a way to finish the proof of Conjecture 2 above using that fact. When I get a chance, I will take a look at the other papers. – Ben W May 14 '16 at 21:34
• Ah, nevermind, that idea doesn't work. Conjecture 2 still wide open. – Ben W May 14 '16 at 21:42