The algebra of bounded linear operators on $\ell_p\oplus\ell_q$ has   infinitely many closed ideals
  We prove that in the reflexive range $1<p<q<\infty$ the algebra of all bounded linear operators on $\ell_p\oplus\ell_q$ has infinitely many closed ideals. This solves a problem raised by A. Pietsch in his book `Operator ideals'. 
Thomas Schlumprecht and Andr\'as Zs\'ak
math.FA