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...

A mean ergodic theorem is proved for a compact operator on a Banach space without assuming mean-boundedness. Some related results are also presented.

In this paper we investigate linear operators between arbitrary BK spaces $X$ and spaces $Y$ of sequences that are $(\overline{N},q)$ summable or bounded. We give necessary and sufficient conditions for infinite matrices $A$ to map $X$ into $Y$. Further, the Hausdorff measure of noncompactness is applied to give necessary and sufficient conditions for $A$ to be a compact operator.

Given Banach spaces  $X$, $Y$ and a compact Hausdorff space  $K$, we use polymeasures to give necessary conditions for a multilinear operator from $C(K,X)$ into  $Y$ to be completely continuous (resp.  unconditionally converging). We deduce necessary and sufficient conditions for  $X$ to have the Schur property (resp.  to contain no copy of  $c_0$), and for  $K$ to be scattered. This extends results concerning linear operators.

We characterize Banach lattices under which each b-weakly compact (resp. b-AM-compact, strong type (B)) operator is L-weakly compact (resp. M-weakly compact).