The Lotz Property in Weakly Compactly Generated Spaces
Abstract
We study whether every infinite-dimensional weakly compactly generated (WCG) Banach space fails the Lotz property. The answer is affirmative in the nonseparable case: a classical decomposition theorem for WCG spaces provides a Schauder decomposition, and an explicit diagonal C0-semigroup has an unbounded generator. The unrestricted assertion is false. Our main abstract result is a Calkin-algebra criterion: if X fails the bounded compact approximation property and, for some fixed m, every noninvertible element of Cal(X)=Lop(X)/Kop(X) has mth power zero, then X has the Lotz property. In particular, this applies when every operator is scalar-plus-strictly-singular and one fixed power of every strictly singular operator is compact. Combining this criterion with the Maurey--Pisier--Szankowski extraction theorem and constructions of Argyros and Motakis yields separable hereditarily indecomposable Lotz spaces over both scalar fields: reflexive examples, and, over the complex field, an example containing no infinite-dimensional reflexive subspace. Moreover, the ambient Argyros--Motakis spaces considered here are Lotz-saturated, although they themselves are non-Lotz. Thus every nonseparable WCG space is non-Lotz, while the separable WCG class contains both Lotz and non-Lotz spaces.
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BibTeX
@misc{araujo2026lotz,
title = {{The Lotz Property in Weakly Compactly Generated Spaces}},
author = {Gustavo Araújo and Mateus Azevedo and Flank Bezerra and Anderson Barbosa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38396},
eprint = {2609.38396},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38396},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Lotz Property in Weakly Compactly Generated Spaces
AU - Gustavo Araújo
AU - Mateus Azevedo
AU - Flank Bezerra
AU - Anderson Barbosa
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38396
UR - https://arxiv.org/abs/2609.38396
AB - We study whether every infinite-dimensional weakly compactly generated (WCG) Banach space fails the Lotz property. The answer is affirmative in the nonseparable case: a classical decomposition theorem for WCG spaces provides a Schauder decomposition, and an explicit diagonal C_0-semigroup has an unbounded generator. The unrestricted assertion is false. Our main abstract result is a Calkin-algebra criterion: if X fails the bounded compact approximation property and, for some fixed m, every noninvertible element of Cal(X)=Lop(X)/Kop(X) has mth power zero, then X has the Lotz property. In particular, this applies when every operator is scalar-plus-strictly-singular and one fixed power of every strictly singular operator is compact. Combining this criterion with the Maurey--Pisier--Szankowski extraction theorem and constructions of Argyros and Motakis yields separable hereditarily indecomposable Lotz spaces over both scalar fields: reflexive examples, and, over the complex field, an example containing no infinite-dimensional reflexive subspace. Moreover, the ambient Argyros--Motakis spaces considered here are Lotz-saturated, although they themselves are non-Lotz. Thus every nonseparable WCG space is non-Lotz, while the separable WCG class contains both Lotz and non-Lotz spaces.
ER -