Departmental Colloquium
Thomas Schlumprecht
Texas A&M
The algebra of bounded linear operators on $\ell_p\oplus\ell_q$, $1
Abstract: For a Banach space $X$ we consider $\mathcal L(X)$, the algebra of linear bounded operators on $X$.
A closed subideal of $\mathcal L(X)$, is a subideal which is closed in the operator norm.
For very few Banach spaces $X$ the structure of the closed subideals of $\mathcal L(X)$ is well
understood.
For example it is known for a long time that the only non trivial closed subideals of
$\mathcal L(\ell_p)$ (other than the zero ideal and the entire algebra) is the ideal of compact operators.
In his book ``Operator Ideals'' Albrecht Pietsch asked about the structure of the closed subideals of
$\mathcal L(\ell_p\oplus\ell_q)$, the space of operators on the complemented sum of $\ell_p$ and $\ell_q$,
where $1\leqslant p < q\leqslant \infty$.
In particular he asked if there are infinitely many closed subideals.
This question was recently solved affirmatively for the reflexive range $1 < p < q < \infty$,
in a joint work by the author in collaboration with Andras Zsak.
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