Maps Preserving Complementarity of Closed Subspaces of a Hilbert Space

Plevnik, Lucijan ; Šemrl, Peter

Canadian journal of mathematics, Tome 66 (2014) no. 5, pp. 1143-1166

Résumé

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.

Détail

DOI : 10.4153/CJM-2013-025-4

Mots-clés : 46B20, 47B49, Hilbert space, lattice of closed subspaces, complemented subspaces, adjacent subspaces, idempotents

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