Similarities of subspace lattices in Banach spaces

Bračič, Janko (author), Kandić, Marko (author)

A collineation of a subspace lattice ${\mathfrak L}$ in a complex Banach space ${\mathscr X}$ is an invertible operator $S$ on ${\mathscr X}$ with the property that the image $S{\mathscr M}$ of a subspace ${\mathscr M}$ belongs to ${\mathfrak L}$ if and and only if ${\mathscr M}$ belongs to it. Hence, $S$ is a collineation of ${\mathfrak L}$ if and only if it implements an order automorphism of ${\mathfrak L}$. We study the group ${\rm Col}({\mathfrak L})$ of all collineations of ${\mathfrak L}$ and its subgroup ${\rm Grp}({\rm Alg}({\mathfrak L}))$ of all invertible operators that fix every subspace in ${\mathfrak L}$. We show that ${\rm Grp}({\rm Alg}({\mathfrak L}))$ is a normal subgroup of ${\rm Col}({\mathfrak L})$; moreover, if ${\mathfrak L}$ is a reflexive subspace lattice, then ${\rm Col}({\mathfrak L})$ is the normalizer of ${\rm Grp}({\rm Alg}({\mathfrak L}))$ in the group of all invertible operators on ${\mathscr X}$. One of the main questions that we consider is whether ${\rm Grp}({\rm Alg}({\mathfrak L}))$ is a complemented subgroup in ${\rm Col}({\mathfrak L})$. For certain subspace lattices ${\mathfrak L}$, such as some realizations of the diamond or the double triangle, some nests in the space of continuous functions on $[0,1]$, and the classical Volterra nest in $L^1[0,1]$, we characterize the complement of ${\rm Grp}({\rm Alg}({\mathfrak L}))$ in ${\rm Col}({\mathfrak L})$. On the other hand, for the Volterra nests in $L^p[0,1]$, where $1<p<\infty$, a further study is needed, and we prove only some partial results.

• Language: English
• Keywords: subspace lattice, collineation, normalizer, reflexive lattice, Volterra nest
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: NTF - Faculty of Natural Sciences and Engineering; FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.12.2026
• Year: 2026
• Number of pages: 20 str.
• Numbering: Vol. 564, iss. 1, art. 130857
• PID: 20.500.12556/RUL-183211
• UDC: 517.9
• ISSN on article: 0022-247X
• DOI: 10.1016/j.jmaa.2026.130857
• COBISS.SI-ID: 280774403
• Publication date in RUL: 08.06.2026

Record is a part of a journal

• Title: Journal of mathematical analysis and applications
• Shortened title: J. math. anal. appl.
• Publisher: Elsevier
• ISSN: 0022-247X
• COBISS.SI-ID: 3081231

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: Banachovi prostori, mreže