The algebra of bounded linear operators on ${\mathrm{\ell }}_p\oplus {\mathrm{\ell }}_q$ has infinitely many closed ideals

We prove that in the reflexive range $1<p<q<\mathrm{\infty }$ the algebra of all bounded linear operators on ${\mathrm{\ell }}_p\oplus {\mathrm{\ell }}_q$ has infinitely many closed ideals. This solves a problem raised by A. Pietsch in his book `Operator ideals’.