Closed ideals of operators acting on some families of sequence spaces

TL;DR

This paper investigates the structure of closed ideals in operator algebras on specific sequence spaces, revealing their isomorphisms to classical spaces and exploring their lattice properties and complementability.

Contribution

It establishes isomorphisms between operator ideals on Tandori and Cesàro sequence spaces and classical sequence spaces, and analyzes the lattice structure and complementability of these spaces.

Findings

Tandori and Cesàro sequence spaces are isomorphic to classical direct sum spaces.

Tandori spaces are complemented in certain Lorentz sequence spaces.

The lattice of closed ideals in some Lorentz and Garling spaces has infinite cardinality.

Abstract

We study the lattice of closed ideals in the algebra of continuous linear operators acting on $p$th Tandori and $p'$th Ces\`{a}ro sequence spaces, $1\leqslant p<\infty$, which we show are isomorphic to the classical sequence spaces $(\oplus_{n=1}^\infty\ell_\infty^n)_p$ and $(\oplus_{n=1}^\infty\ell_1^n)_{p'}$, respectively. We also show that Tandori sequence spaces are complemented in certain Lorentz sequence spaces, and that the lattice of closed ideals for certain other Lorentz and Garling sequence spaces has infinite cardinality.