Maps preserving complementarity of closed subspaces of a Hilbert space (Q2922906)

Let \({\mathcal H}\) and \({\mathcal K}\) be two separable infinite-dimensional real/complex Hilbert spaces, and let \({\mathcal I}({\mathcal H})\) be the set of bounded linear idempotent operators on \({\mathcal H}\). Let \(\text{Lat}({\mathcal H})\) be the collection of all closed subspaces of \({\mathcal H}\), and NEWLINE\[NEWLINE{\mathcal C}_{{\mathcal H}}:=\{\{U,V\}:U,V\in\text{Lat}({\mathcal H}),~{\mathcal H}=U\oplus V \}NEWLINE\]NEWLINE be the collection of closed subspaces of \({\mathcal H}\) that are complemented. Among other results in this interesting paper, the authors first characterize pairs of bijective maps \(\phi\) and \(\psi\) from \(\text{Lat}({\mathcal H})\) into \(\text{Lat}({\mathcal K})\) that preserve complementarity of subspaces; i.e., NEWLINE\[NEWLINE\{U,~V\}\in{\mathcal C}_{{\mathcal H}}\iff \{\phi(U),\psi(V)\}\in{\mathcal C}_{{\mathcal K}},\;U,V\in\text{Lat}({\mathcal H}).NEWLINE\]NEWLINE Secondly, they obtain a complete description of bijective maps from \({\mathcal I}({\mathcal H})\) into \({\mathcal I}({\mathcal K})\) preserving the equality of images and the equality of kernels. Furthermore, they show that several structural results for maps on idempotent operators can be deduced from such a description.