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It is well known that the only proper non-trivial norm closed ideal in the algebra L(X) for $X={\ell }_p$ (1 ≤ p < ∞) or X = c₀ is the ideal of compact operators. The next natural question is to describe all closed ideals of $L\left({\ell }_p\oplus {\ell }_q\right)$ for 1 ≤ p,q < ∞, p ≠ q, or equivalently, the closed ideals in $L({\ell }_p,{\ell }_q)$ for p < q. This paper shows that for 1 < p < 2 < q < ∞ 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...