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) $K(\overline{B})$ 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 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 ithe 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?

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?

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) $K(\overline{B})$ 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. Question 1. Is the interpolated operator $T_{JK}:J(\overline{A})\rightarrow K(\overline{B})$ weakly compact if and only if the 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?. Question 2. What about other operator ideals such as Rosenthal, Decomposing (or Asplund), Banach-Saks and Alternate sign Banach-Saks operator ideals?

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) $K(\overline{B})$ 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 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 ithe 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?

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) $K(\overline{B})$ 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. Is the interpolated operator $T_{JK}:J(\overline{A})\rightarrow K(\overline{B})$ weakly compact if and only if the 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?. What about other real methods for finite families and operator ideals such as Rosenthal, Banach-Saks, Alternate sign Banach-Saks and many other?

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) $K(\overline{B})$ 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. Question 1. Is the interpolated operator $T_{JK}:J(\overline{A})\rightarrow K(\overline{B})$ weakly compact if and only if the 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?. Question 2. What about other operator ideals such as Rosenthal, Decomposing (or Asplund), Banach-Saks and Alternate sign Banach-Saks operator ideals?