Closed ideals in $\mathcal{L}(X)$ and $\mathcal{L}(X^{\ast })$ when $X$ contains certain copies of ${\ell }_p$ and $c_0$

Abstract

Suppose $X$ is a real or complexified Banach space containing a complemented copy of ${\ell }_p$, $p\in (1,2)$, and a copy (not necessarily complemented) of either ${\ell }_q$, $q\in (p,\infty )$, or $c_0$. Then $\mathcal{L}(X)$ and $\mathcal{L}(X^{\ast })$ each admit continuum many closed ideals. If in addition $q\ge p^{\prime }$, $\frac{1}{p}+\frac{1}{p^{\prime }}=1$, then the closed ideals of $\mathcal{L}(X)$ and $\mathcal{L}(X^{\ast })$ each fail to be linearly ordered. We obtain additional results in the special cases of $\mathcal{L}({\ell }_1\oplus {\ell }_q)$ and $\mathcal{L}({\ell }_p\oplus c_0)$, $1<p<2<q<\infty$.

28 pages

References in corpus (1)

• The algebra of bounded linear operators on ${\ell }_p\oplus {\ell }_q$ has infinitely many closed ideals