Regular lattices of tensor products (Q1932617)

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.