Regular lattices of tensor products
type: Item
instance of: scholarly article
title: Regular lattices of tensor products
zbMATH Open document ID: 1272.47033
published in: Linear Algebra and its Applications
publication date: 2013-01-21T00:00:00Z
review text: The author investigates invariant subspaces of tensor products of operators on Hilbert spaces. There are considered nonzero Hilbert spaces (either finite- or infinite-dimensional, not necessarily separable). Denote by \({\mathcal B}({\mathcal H})\) the set of all linear bounded operators on Hilbert space \({\mathcal H}\). For \(A\in {\mathcal B}({\mathcal H})\), denote by \(\text{Lat}(A)\) the lattice of all \(A\)-invariant subspaces of \({\mathcal H}\). Consider the finite collections of Hilbert spaces \(\{ {\mathcal H}_{i}\}_{i=1}^{m}\) and operators \(\{ A_{i}\}_{i=1}^{m}\) with each \(A_{i}\in{\mathcal B}({\mathcal H}_{i})\). Let \(\bigotimes_{i=1}^{m}A_{i}\) be the operator on the Hilbert space \(\bigotimes_{i=1}^{m}{\mathcal H}_{i}\), and consider \[ \bigotimes_{i=1}^{m}\text{Lat}(A_{i}) = \left\{ \{0\}\neq\bigotimes_{i=1}^{m}{\mathcal M}_{i} \subset \bigotimes_{i=1}^{m}{\mathcal H}_{i}: \{0\}\neq{\mathcal M}_{i}\in\text{Lat}(A_{i}) \right\}, \] \[ \text{RLat} \left( \bigotimes_{i=1}^{m}A_{i} \right) = \left\{ \bigotimes_{i=1}^{m}{\mathcal M}_{i} \subset \bigotimes_{i=1}^{m}{\mathcal H_{i}}: \bigotimes_{i=1}^{m}{\mathcal M}_{i} \in \text{Lat} \left( \bigotimes_{i=1}^{m}A_{i} \right) \right\}. \] The main result of the paper goes as follows. Theorem. \( \text{RLat} (\bigotimes_{i=1}^{m}A_{i}) \) is a lattice. If each \(A_{i}\) is injective, then \[ \text{RLat} \left( \bigotimes_{i=1}^{m}A_{i} \right) \setminus \{0\} = \bigotimes_{i=1}^{m}\text{Lat}(A_{i}) \subset \text{Lat} \left( \bigotimes_{i=1}^{m}A_{i} \right) \setminus \{0\}, \] and the inclusion may be proper.
reviewed by: Vladimir S. Pilidi
Mathematics Subject Classification ID: 47A80
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A46
Mathematics Subject Classification ID: 47A15
zbMATH DE Number: 6127352
zbMATH Keywords: tensor product
zbMATH Keywords: Hilbert space
zbMATH Keywords: invariant subspace
author: Carlos S. Kubrusly
MaRDI profile type: Publication
full work available at URL: j.laa.2012.07.013
OpenAlex ID: W2136572268
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DOI: 10.1016/J.LAA.2012.07.013
description: scientific article; zbMATH DE number 6127352