We show for $2\le p<\infty$ and subspaces $X$ of quotients of $L_p$ with a $1$-unconditional finite-dimensional Schauder decomposition that $K(X,{\ell }_p)$ is an $M$-ideal in $L(X,{\ell }_p)$.

We study the position of compact operators in the space of all continuous linear operators and its subspaces in terms of ideals. One of our main results states that for Banach spaces $X$ and $Y$ the subspace of all compact operators $𝒦(X,Y)$ is an $M(r_1{r}_2,s_1{s}_2)$-ideal in the space of all continuous linear operators $ℒ(X,Y)$ whenever $𝒦(X,X)$ and $𝒦(Y,Y)$ are $M(r_1,s_1)$- and $M(r_2,s_2)$-ideals in $ℒ(X,X)$ and $ℒ(Y,Y)$, respectively, with $r_1+s_1/2>1$ and $r_2+s_2/2>1$. We also prove that the $M(r,s)$-ideal $𝒦(X,Y)$ in $ℒ(X,Y)$ is separably determined. Among others, our results complete and improve some well-known results...

Nigel J. Kalton was one of the most eminent guests participating in the Józef Marcinkiewicz Centenary Conference. His contribution to the scientific aspect of the meeting was very essential. Nigel was going to prepare a paper based on his plenary lecture. The editors are completely sure that the paper would be a real ornament of the Proceedings. Unfortunately, Nigel's sudden death totally destroyed editors' hopes and plans. Every mathematician knows how unique were Nigel's mathematical achievements....