  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
$\mathcal{L}(X^*)$ each admit continuum many closed ideals. If in addition
$q\geq p'$, $\frac{1}{p}+\frac{1}{p'}=1$, then the closed ideals of
$\mathcal{L}(X)$ and $\mathcal{L}(X^*)$ each fail to be linearly ordered. We
obtain additional results in the special cases of
$\mathcal{L}(\ell_1\oplus\ell_q)$ and $\mathcal{L}(\ell_p\oplus c_0)$,
$1<p<2<q<\infty$.
