On norm closed ideals in $L(\ell _p,\ell _q)$
Volume 179 / 2007
Studia Mathematica 179 (2007), 239-262
MSC: Primary 47L20; Secondary 47B10, 47B37.
DOI: 10.4064/sm179-3-3
Abstract
It is well known that the only proper non-trivial norm closed ideal in the algebra $L(X)$ for $X=\ell _p$ $(1\le p< \infty )$ or $X=c_0$ is the ideal of compact operators. The next natural question is to describe all closed ideals of $L(\ell _p\oplus \ell _q)$ for $1\le p,q< \infty $, $p\not =q$, or equivalently, the closed ideals in $L(\ell _p,\ell _q)$ for $p< q$. This paper shows that for $1< p< 2< q< \infty $ there are at least four distinct proper closed ideals in $L(\ell _p,\ell _q)$, including one that has not been studied before. The proofs use various methods from Banach space theory.
