Geodesic
Maps Preserving Complementarity of Closed Subspaces of a Hilbert Space
Canadian journal of mathematics, Tome 66 (2014) no. 5, pp. 1143-1166

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DOI

Let $\mathcal{H}$ and $\mathcal{K}$ be infinite-dimensional separable Hilbert spaces and $\text{Lat}\,\mathcal{H}$ the lattice of allclosed subspaces oh $\mathcal{H}$ . We describe the general form of pairs of bijective maps $\phi ,\,\psi :\,\text{Lat}\,\mathcal{H}\,\to \,\text{Lat}\,\mathcal{K}$ having the property that for every pair $U,\,V\,\in \,\text{Lat}\,\mathcal{H}$ we have $\mathcal{H}\,=\,U\,\oplus \,V\,\Leftrightarrow \,\mathcal{K}\,=\,\phi \left( U \right)\,\oplus \,\psi \,\left( V \right)$ . Then we reformulate this theorem as a description of bijective image equality and kernel equality preserving maps acting on bounded linear idempotent operators. Several known structural results for maps on idempotents are easy consequences.
DOI : 10.4153/CJM-2013-025-4
Mots-clés : 46B20, 47B49, Hilbert space, lattice of closed subspaces, complemented subspaces, adjacent subspaces, idempotents 
Plevnik, Lucijan; Šemrl, Peter. Maps Preserving Complementarity of Closed Subspaces of a Hilbert Space. Canadian journal of mathematics, Tome 66 (2014) no. 5, pp. 1143-1166. doi: 10.4153/CJM-2013-025-4
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     author = {Plevnik, Lucijan and \v{S}emrl, Peter},
     title = {Maps {Preserving} {Complementarity} of {Closed} {Subspaces} of a {Hilbert} {Space}},
     journal = {Canadian journal of mathematics},
     pages = {1143--1166},
     year = {2014},
     volume = {66},
     number = {5},
     doi = {10.4153/CJM-2013-025-4},
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