The algebra of bounded linear operators on ℓp ⊕ ℓq has infinitely many closed ideals

We prove that in the reflexive range 1 < p < q < ∞, the algebra ℒ(ℓp⊕ℓq) of all bounded linear operators on ℓp⊕ℓq has infinitely many closed ideals. This solves a problem raised by A. Pietsch [Operator ideals, Math. Monogr. 16, VEB Deutscher Verlag der Wissenschaften, Berlin 1978, Problem 5.3.3] in his book `Operator ideals'.The first author’s research was supported by NSF grant DMS-1160633. The second author was supported by the 2014 Workshop in Analysis and Probability at Texas A&M University