On norm closed ideals in
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.
