| Title: | Similarities of subspace lattices in Banach spaces |
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| Authors: | ID Bračič, Janko (Author)
ID Kandić, Marko (Author) |
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| Files: |  PDF - Presentation file, download (1,28 MB)
MD5: EC27BD0D5B28A00385983D587C624D27
 URL - Source URL, visit https://www.sciencedirect.com/science/article/pii/S0022247X26004695
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | 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. |
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| Keywords: | subspace lattice, collineation, normalizer, reflexive lattice, Volterra nest |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.12.2026 |
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| Year of publishing: | 2026 |
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| Number of pages: | 20 str. |
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| Numbering: | Vol. 564, iss. 1, article no. 130857 |
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| PID: | 20.500.12556/DiRROS-29841  |
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| UDC: | 517.9 |
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| ISSN on article: | 0022-247X |
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| DOI: | 10.1016/j.jmaa.2026.130857 |
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| COBISS.SI-ID: | 280774403 |
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| Note: |
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| Publication date in DiRROS: | 08.06.2026 |
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| Views: | 64 |
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| Downloads: | 40 |
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