Geodesic
Remarks on Invariant Subspace Lattices
Canadian mathematical bulletin, Tome 12 (1969) no. 5, pp. 639-643

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If A is a bounded linear operator on an infinite-dimensional complex Hilbert space H, let lat A denote the collection of all subspaces of H that are invariant under A; i.e., all closed linear subspaces M such that x ∈ M implies (Ax) ∈ M. There is very little known about the question: which families F of subspaces are invariant subspace lattices in the sense that they satisfy F = lat A for some A? (See [5] for a summary of most of what is known in answer to this question.) Clearly, if F is an invariant subspace lattice, then {0} ∈ F, H ∈ F and F is closed under arbitrary intersections and spans. Thus, every invariant subspace lattice is a complete lattice. 
Rosenthal, Peter. Remarks on Invariant Subspace Lattices. Canadian mathematical bulletin, Tome 12 (1969) no. 5, pp. 639-643. doi: 10.4153/CMB-1969-082-1
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     author = {Rosenthal, Peter},
     title = {Remarks on {Invariant} {Subspace} {Lattices}},
     journal = {Canadian mathematical bulletin},
     pages = {639--643},
     year = {1969},
     volume = {12},
     number = {5},
     doi = {10.4153/CMB-1969-082-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1969-082-1/}
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