The algebra of bounded linear operators on $\ell_p\oplus\ell_q$ has infinitely many closed ideals

TL;DR

This paper proves that the algebra of bounded linear operators on the direct sum of p and q spaces has infinitely many closed ideals when 1<p<q<, resolving a question posed by A. Pietsch.

Contribution

It establishes the existence of infinitely many closed ideals in the operator algebra on pq spaces for the first time.

Findings

The algebra has infinitely many closed ideals for 1<p<q<.

This solves a longstanding open problem in operator theory.

The result applies to reflexive p and q spaces with 1<p<q<.

Abstract

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'.