Let $\overline{A} \mbox{ and } \overline{B}$ be n+1-tuples of Banach spaces and $T:\overline{A}\rightarrow \overline{B}$ be an interpolation operator; let $J(\overline{A})$ and (the corresponding, depending on the same parameters) $K(\overline{B})$ be the interpolation spaces obtained from $\overline{A}$ and $\overline{B}$ by the $J$ and $K$ real methods of interpolation for finite families of G. Sparr, D. L. Fernández or Cobos-Petre.
Q1. If the associated operator $T_{\mathcal{J}\mathcal{S}}:\mathcal{J}(\overline{A}) \rightarrow \mathcal{S}(\overline{B})$ from the intersection space into the sum space is weakly compact, is the interpolated operator $T_{JK}:J(\overline{A})\rightarrow K(\overline{B})$ also weakly compact?
Q2. What about other operator ideals such as Rosenthal, Decomposing (or Asplund), Banach-Saks and Alternate sign Banach-Saks operator ideals?
