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# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## On norm closed ideals in $L(\ell _p,\ell _q)$

### Tom 179 / 2007

Studia Mathematica 179 (2007), 239-262 MSC: Primary 47L20; Secondary 47B10, 47B37. DOI: 10.4064/sm179-3-3

#### Streszczenie

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.

#### Autorzy

• B. SariDepartment of Mathematics
University of North Texas
Denton, TX 76203-1430, U.S.A.
e-mail
• Th. SchlumprechtDepartment of Mathematics
Texas A&M University
College Station, TX 77843-3368, U.S.A.
e-mail
• N. Tomczak-JaegermannDepartment of Mathematical and Statistical Sciences
University of Alberta