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arXiv · 2012.08935

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

Ben Wallis
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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.

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Ben Wallis. 2020-12-16. A new complemented subspace for the Lorentz sequence spaces, with an application to its lattice of closed ideals. https://arxiv.org/abs/2012.08935

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