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We prove that in the reflexive range1  &lt;  p  &lt;  q  &lt;  ∞{1&lt;p&lt;q&lt;\infty}   , the algebraℒ  ⁢   (    ℓ  p   ⊕   ℓ  q    )\mathcal{L}(\ell_{p}\oplus\ell_{q})    of all bounded linear operators onℓ  p   ⊕   ℓ  q{\ell_{p}\oplus\ell_{q}}    has infinitely many closed ideals. This solves a problem raised by A. Pietsch [ 4 , Problem 5.3.3] in his book ‘Operator ideals’.

The algebra of bounded linear operators\break on ℓ p ⊕ ℓ p has infinitely many closed ideals