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A new complemented subspace for the Lorentz sequence spaces, with an application to its lattice of closed ideals

Part of: Linear spaces and algebras of operators Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  02 August 2021

Ben Wallis
Ben Wallis*
Affiliation:
Division of Math/Science/Business, Kishwaukee College, Malta, IL60150, USA
*
e-mail: benwallis@live.com
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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.

Keywords

Lorentz sequence spacescomplemented subspaceslattice of closed ideals

MSC classification

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 46B45: Banach sequence spaces 47L20: Operator ideals
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 3 , September 2022 , pp. 759 - 769
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Copyright
© Canadian Mathematical Society 2021

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