A new complemented subspace for the Lorentz sequence spaces, with an application to its lattice of closed ideals. Issue 3 (2nd September 2022)

• Record Type: Journal Article
• Authors: Wallis, Ben
• Is Part Of: Canadian mathematical bulletin =. Volume 65:Issue 3(2022)
• Page Start: 759
• Page End: 769
• Publication Date: 2022-09-02
• Subjects: 46B20 -- 46B45 -- 47L20; Lorentz sequence spaces -- complemented subspaces -- lattice of closed ideals; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/S0008439521000576
• Languages: English
• ISSNs: 0008-4395

Abstract

Abstract: We show that every Lorentz sequence space $d(\textbf {w}, p)$ admits a 1-complemented subspace Y distinct from $\ell _p$ and containing no isomorph of $d(\textbf {w}, p)$ . In the general case, this is only the second nontrivial complemented subspace in $d(\textbf {w}, p)$ yet known. We also give an explicit representation of Y in the special case $\textbf {w}=(n^{-\theta })_{n=1}^\infty $ ( $0<\theta <1$ ) as the $\ell _p$ -sum of finite-dimensional copies of $d(\textbf {w}, p)$ . As an application, we find a sixth distinct element in the lattice of closed ideals of $\mathcal {L}(d(\textbf {w}, p))$, of which only five were previously known in the general case.