Is $T^{**}$ unconditionally $p$-summing whenever $T$ is unconditionally $p$-summing?

Question

A sequence $(x_{n})_{n}$ in a Banach space $X$ is said to be unconditionally $p$-summable if $$\sup_{x^{*}\in B_{X^{*}}}\Bigl(\sum_{n=m}^{\infty}\lvert\langle x^{*},x_{n}\rangle\rvert^{p}\Bigr)^{1/p}\rightarrow 0\qquad(m\rightarrow \infty).$$ We say that an operator $T:X\rightarrow Y$ is unconditionally $p$-summing if $(Tx_{n})_{n}$ is unconditionally $p$-summable in $Y$ whenever $(x_{n})_{n}$ is weakly $p$-summable in $X$. We can prove that $T$ is unconditionally $p$-summing whenever $T^{**}$ is unconditionally $p$-summing. Is the converse true?

Answer 1

The answer is no. Bourgain and Delbaen constructed a Banach space $X$ that has the Schur property and $X^{**}$ is isomorphically universal for separable Banach space (it is even isomorphic to the the second dual of $C[0,1]$). Every operator from $\ell_p$, $1<p<\infty$, and from $c_0$ into $X$ is thus compact, so that every operator with domain $X$ is unconditionally $p$-summing for all $1\le p < \infty$. But the second adjoint of the identity operator on $X$ is not unconditionally $p$-summing for any $1\le p < \infty$ because it is an isomorphism on copies of $\ell_p$ for all $p$.

Bourgain, J.; Delbaen, F. A class of special $L_\infty$ spaces. Acta Math. 145 (1980), no. 3-4, 155–176.