Thomas Schlumprecht : The algebra of bounded linear operators on $\ell_p\oplus\ell_q$, $1<p<q<\infty$ has infinitely many closed subideals

Posted by Shmuel Friedland , part of the Departmental Colloquium.

At

April 15, 2016, 3 p.m.

In

SEO 636

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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