An operator ideal $\mathcal{I}$ possesses the $\sum_{p}$-property, say $1<p<\infty$, if for arbitrary collections of Banach spaces $E_{m}, F_{n} (m,n=1,2,\ldots)$ the following holds:

if $T\in \mathcal{L}((\sum_{m} E_{m})_{p},(\sum_{n} F_{n})_{p})$ and $Q_{n}TJ_{m} \in \mathcal{I}(E_{m},F_{n}) \mbox { for every } m,n$ then $T\in \mathcal{I}((\sum_{m} E_{m})_{p},(\sum_{n} F_{n})_{p})$

($J_{m}, Q_{n}$ are the natural injections and projections), see [1, page 404].

Many things are known, see [1]. For example:

i) if the ideal has the $\sum_{p}$-property then it is closed.

ii) the operator ideals of weakly compact operators, Rosenthal, separable, Banach-Saks, Alternating sign Banach-Saks, Decomposing and others have the $\sum_{p}$-property.

iii) also are known some conditions that imply the $\sum_{p}$-property.

$\sum_{p}$-property is called p-stability by Piestch in his book, see [2, Epilogue].

Question 1: Does $\mathcal{I}$ closed and $\ell_{p}\in$ space($\mathcal{I}$) (that is, identity $1_{\ell_{p}}\in\mathcal{I}$) imply $\mathcal{I} \mbox{ has }\sum_{p}$-property?.

Question 2: If the answer to the above question is negative, does $\mathcal{I}$ closed + $\ell_{p}\in$ space($\mathcal{I}$) + something imply $\mathcal{I} \mbox{ has }\sum_{p}$-property?.

Question 3: Does someone know of a book or paper where this concept of $\sum_{p}$-property for ideals is studied?.

[1] S. Heinrich, Closed operator ideals and interpolation, J. Funct. Anal. $\mathbf{35}$ (1980), 397-411.

[2] A. Pietsch, Operator ideals, North Holland, Amsterdam, 1980.