  It is well known that the only proper non-trivial norm-closed ideal in the
algebra L(X) for X=\ell_p (1 \le p < \infty) or X=c_0 is the ideal of compact
operators. The next natural question is to describe all closed ideals of
L(\ell_p\oplus\ell_q) for 1 \le p,q < \infty, p \neq q, or, equivalently, the
closed ideals in L(\ell_p,\ell_q) for p < q. This paper shows that for 1 < p <
2 < q < \infty there are at least four distinct proper closed ideals in
L(\ell_p,\ell_q), including one that has not been studied before. The proofs
use various methods from Banach space theory.
