Let $X,Y$ be Banach spaces. We denote by $\mathcal{L}(X,Y)$ the space of all operators from $X$ into $Y$, $\mathcal{K}(X,Y)$ by the space of all the compact operators from $X$ into $Y$, $S(X,Y)$ by the space of all strictly singular operators from $X$ to $Y$ and $SC(X,Y)$ by the space of all strictly cosingular operators from $X$ to $Y$.

It was proved in Theorem 2.4.10 [F. Albiac and N. Kalton, Topics in Banach space theory] that for every Banach space $Y$, one has $\mathcal{K}(c_{0}, Y)=S(c_{0},Y)$.

Definition 1. We say that a Banach space $X$ has the Bessaga-Pełczyński property I (this name is used temporarily here) if $\mathcal{K}(X,Y)=S(X,Y)$ for every Banach space $Y$.

Question 1. Is there any Banach space enjoying the Bessaga-Pełczyński property I besides $c_{0}$?

Question 2. Is the Bessaga-Pełczyński property I interesting?

Definition 2. We say that a Banach space $Y$ has the Bessaga-Pełczyński property II if $\mathcal{K}(X,Y)=SC(X,Y)$ for every Banach space $X$.

It was proved by Pełczyński in 1965 that $l_{1}$ has the Bessaga-Pełczyński property II.

Question 3. Is there any Banach space enjoying the Bessaga-Pełczyński property II besides $l_{1}$?

Thank you!