$M(r,s)$-ideals of compact operators

Haller, Rainis ; Johanson, Marje ; Oja, Eve

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 3, pp. 673-693.

Résumé

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 $\mathcal{K}(X,Y)$ is an $M(r_1{r}_2,s_1{s}_2)$-ideal in the space of all continuous linear operators $\mathcal{L}(X,Y)$ whenever $\mathcal{K}(X,X)$ and $\mathcal{K}(Y,Y)$ are $M(r_1,s_1)$- and $M(r_2,s_2)$-ideals in $\mathcal{L}(X,X)$ and $\mathcal{L}(Y,Y)$, respectively, with $r_1+s_1/2>1$ and $r_2+s_2/2>1$. We also prove that the $M(r,s)$-ideal $\mathcal{K}(X,Y)$ in $\mathcal{L}(X,Y)$ is separably determined. Among others, our results complete and improve some well-known results on $M$-ideals.

Détail

DOI : 10.1007/s10587-012-0059-9

Classification : 46B04, 46B20, 46B28, 47L05
Mots-clés : $M(r,s)$-ideal and $M$-ideal of compact operators; property $M^{\ast }(r,s)$; compact approximation property